Both tuberculosis (TB) and COVID-19 are infectious diseases with similar clinical manifestations, which mainly affect the lungs. Clinical studies have revealed that the immunosuppressive drugs taken by COVID-19 patients can affect the immunological functions in the body, which can cause the patients to contract active TB via a new infection or reinfection, and the co-infection of the two diseases portends a clinical complexity in the management of the patients. Thus, this paper presents a mathematical model to study the dynamics and control of COVID-19-TB co-infection. The full model of the co-infection is split into two submodels, namely, the TB-only and the COVID-19-only models. The equilibria of the disease-free and endemic situations of the two sub-models are shown to be globally asymptotically stable when their control reproduction numbers RTVo,RCVo<1 and ˜RTVo,˜RCVo>1, respectively. However, the disease-free equilibrium of the co-infection model was found to lose its global stability property when the reproduction number RFo<1, therefore exhibiting a backward bifurcation. Uncertainty and sensitivity analysis of the associated reproduction number of the full model has been performed by using the Latin hypercube sampling/Pearson rank correlation coefficient (LHS/PRCC) method. The rate of transmission of COVID-19 and the proportions of individuals vaccinated with Bacillus Calmette-Guérin (BCG) and against COVID-19 were found to be highly significant in the spread and control of COVID-19-TB co-infection. Furthermore, the simulation results show that decreasing the COVID-19 transmission rate and increasing the proportion of people vaccinated with BCG and against COVID-19 can lower the number of cases of COVID-19-TB co-infection. Therefore, measures to reduce the transmission rate and the provision of adequate resources to increase the proportions of people vaccinated against TB and COVID-19 should be implemented to minimize the cases of co-infection.
Citation: J. O. Akanni, S. Ajao, S. F. Abimbade, Fatmawati. Dynamical analysis of COVID-19 and tuberculosis co-infection using mathematical modelling approach[J]. Mathematical Modelling and Control, 2024, 4(2): 208-229. doi: 10.3934/mmc.2024018
[1] | Andrey Muravnik . Wiener Tauberian theorem and half-space problems for parabolic and elliptic equations. AIMS Mathematics, 2024, 9(4): 8174-8191. doi: 10.3934/math.2024397 |
[2] | Zhanwei Gou, Jincheng Shi . Blow-up phenomena and global existence for nonlinear parabolic problems under nonlinear boundary conditions. AIMS Mathematics, 2023, 8(5): 11822-11836. doi: 10.3934/math.2023598 |
[3] | Apassara Suechoei, Parinya Sa Ngiamsunthorn . Extremal solutions of φ−Caputo fractional evolution equations involving integral kernels. AIMS Mathematics, 2021, 6(5): 4734-4757. doi: 10.3934/math.2021278 |
[4] | Hao-Yue Liu, Wei Zhang . Neumann gradient estimate for nonlinear heat equation under integral Ricci curvature bounds. AIMS Mathematics, 2024, 9(2): 3881-3894. doi: 10.3934/math.2024191 |
[5] | Yanyan Cui, Chaojun Wang . Dirichlet and Neumann boundary value problems for bi-polyanalytic functions on the bicylinder. AIMS Mathematics, 2025, 10(3): 4792-4818. doi: 10.3934/math.2025220 |
[6] | Nataliya Gol’dman . Nonlinear boundary value problems for a parabolic equation with an unknown source function. AIMS Mathematics, 2019, 4(5): 1508-1522. doi: 10.3934/math.2019.5.1508 |
[7] | Yonghui Zou, Xin Xu, An Gao . Local well-posedness to the thermal boundary layer equations in Sobolev space. AIMS Mathematics, 2023, 8(4): 9933-9964. doi: 10.3934/math.2023503 |
[8] | Hongmei Li . Convergence of smooth solutions to parabolic equations with an oblique derivative boundary condition. AIMS Mathematics, 2024, 9(2): 2824-2853. doi: 10.3934/math.2024140 |
[9] | Heping Jiang . Complex dynamics induced by harvesting rate and delay in a diffusive Leslie-Gower predator-prey model. AIMS Mathematics, 2023, 8(9): 20718-20730. doi: 10.3934/math.20231056 |
[10] | Fan Wan, Xiping Liu, Mei Jia . Ulam-Hyers stability for conformable fractional integro-differential impulsive equations with the antiperiodic boundary conditions. AIMS Mathematics, 2022, 7(4): 6066-6083. doi: 10.3934/math.2022338 |
Both tuberculosis (TB) and COVID-19 are infectious diseases with similar clinical manifestations, which mainly affect the lungs. Clinical studies have revealed that the immunosuppressive drugs taken by COVID-19 patients can affect the immunological functions in the body, which can cause the patients to contract active TB via a new infection or reinfection, and the co-infection of the two diseases portends a clinical complexity in the management of the patients. Thus, this paper presents a mathematical model to study the dynamics and control of COVID-19-TB co-infection. The full model of the co-infection is split into two submodels, namely, the TB-only and the COVID-19-only models. The equilibria of the disease-free and endemic situations of the two sub-models are shown to be globally asymptotically stable when their control reproduction numbers RTVo,RCVo<1 and ˜RTVo,˜RCVo>1, respectively. However, the disease-free equilibrium of the co-infection model was found to lose its global stability property when the reproduction number RFo<1, therefore exhibiting a backward bifurcation. Uncertainty and sensitivity analysis of the associated reproduction number of the full model has been performed by using the Latin hypercube sampling/Pearson rank correlation coefficient (LHS/PRCC) method. The rate of transmission of COVID-19 and the proportions of individuals vaccinated with Bacillus Calmette-Guérin (BCG) and against COVID-19 were found to be highly significant in the spread and control of COVID-19-TB co-infection. Furthermore, the simulation results show that decreasing the COVID-19 transmission rate and increasing the proportion of people vaccinated with BCG and against COVID-19 can lower the number of cases of COVID-19-TB co-infection. Therefore, measures to reduce the transmission rate and the provision of adequate resources to increase the proportions of people vaccinated against TB and COVID-19 should be implemented to minimize the cases of co-infection.
In this paper, we study the existence, uniqueness, decay estimates, and the large-time behavior of the solutions for a class of the nonlinear strongly degenerate parabolic equations involving the linear inhomogeneous heat equation solution as a source term under Neumann boundary conditions with bounded Radon measure as initial data. This problem is described as follows:
{ut=Δψ(u)+h(t)f(x,t) in Q:=Ω×(0,T),∂ψ(u)∂η=g(u) on S:=∂Ω×(0,T),u(x,0)=u0(x) in Ω, | (P) |
where T>0, Ω⊂RN(N≥2) is an open bounded domain with smooth boundary ∂Ω, η is an unit outward normal vector. The initial value data u0 is a nonnegative bounded Radon measure on Ω. The functions ψ and g fulfill the following assumptions
{(i)ψ∈L∞(R+)∩C2(R+), ψ(0)=0, ψ′>0 in R+,(ii)ψ′,ψ″∈L∞(R+) and ψ′(s)→0ass→+∞,(iii)ψ(s)→γass→+∞,(iv)∣ψ″∣ψ′≤κinR∗+,for someκ∈R∗+, | (I) |
and
{(i)g∈L∞(R+)∩C1(R+), g′<0 in R+ and g>0 in R+,(ii)g′∈L∞(R+)andg(s)→0ass→+∞, | (A) |
where R+≡[0,+∞), R∗+≡(0,+∞) and γ∈R∗+. By ψ′ and ψ″ we denote the first and second derivatives of the function ψ. The assumption (I)-(iii) stem from (I)-(i), hence we extend the function ψ in [0,+∞] defining ψ(+∞)=γ.
The typical example of the functions ψ and g are given
ψ(s)=γ[1−e1−(1+s)m]andg(s)=e1−(1+s)m. | (1.1) |
where 0<m≤1.
The function h satisfies the following hypothesis
h∈C1(R+)∩L1(R+),h(0)=0,h′>0inR+. | (J) |
The function f is a solution of the linear inhomogeneous heat equation under Neumann boundary conditions with measure data
{ft=Δf+μ in Q:=Ω×(0,T),∂f∂η=g(f) on S:=∂Ω×(0,T),f(x,0)=u0(x) in Ω, | (H) |
where μ is a nonnegative bounded Radon measure on Q and g fulfills the assumption (A).
Throughout this paper, we consider solutions of the problem (P) as maps from (0, T) to the cone of nonnegative finite Radon measure on Ω, which satisfy (P) in the following sense: For a suitable class of test functions ξ there holds
∫T0⟨ur(⋅,t),ξt(⋅,t)⟩Ωdt+∫T0h(t)⟨f(⋅,t),ξ(⋅,t)⟩Ωdt+⟨u0,ξ(⋅,0)⟩+ |
+∫T0⟨g(ur(⋅,t)),ξ⟩∂Ωdt=∫T0⟨∇ψ(ur)(⋅,t),∇ξ(⋅,t)⟩Ωdt | (1.2) |
(see Definition 2.1). Here the measure u(⋅,t) is defined for almost every t∈(0,T), ur∈L1(Q).
The type of the problem (P) has been intensively studied by many authors for instance (see [5,18,19,20,27,28,30]) few to mention. For the general form of the problem (P), we consider the following problem studied in [18],
{ut=div(∇ϕ(x,t,u)+h(x,t,u))+F(x,t,u) in ΩT,(∇ϕ(x,t,u)+h(x,t,u))⋅η=r(x,t,u) on ∑T,u(x,0)=u0 in (∂Ω∖∑)T∪¯Ω×{0}, | (A.1) |
where ΩT=Ω×(0,T), ∑T=∑×(0,T), (∂Ω∖∑)T=(∂Ω∖∑)×(0,T) with ∑ is a relative open subset of ∂Ω, ¯∑ and ∂Ω∖∑ are C2 surface with boundary which meet in C2 manifold dimension N−2 and 0≤u0∈L∞((∂Ω∖∑)T∪¯Ω×{0}). The author in [18], proved the local existence, uniqueness and the blow-up at the finite time of the degenerate parabolic equations (A.1). Furthermore, the existence and regularity of the solutions to the quasilinear parabolic systems under nonlinear boundary conditions is discussed in detail by the studies [28,29]
{ut+A(t,u)u=F(t,u) in Ω×(s,T),β(t,u)u=r(t,u) in ∂Ω×(s,T],u(s)=u0 on Ω, | (A.2) |
where s<t≤T and u0∈Wτ,p(Ω,RN)(τ∈[0,∞)) and the definition of the operators A(t,u)u and β(t,u)u are in [28]. Similarly, studies in [19,20] showed the existence and regularity of the degenerate parabolic equations with nonlinear boundary conditions and u0∈L2(Ω) as an initial datum. Thus, we point out that the difference between the previous works (A.1), (A.2) and our work is on the following points; firstly, the initial value u0∈M+(Ω) (the nonnegative bounded Radon measure on Ω), secondly, the assumptions of the functions ψ, g given by (I) and (A). Finally, the source term f is a solution to the linear inhomogeneous heat equation under Neumann boundary conditions with measure data.
Furthermore, the study of the degenerate parabolic problem with forcing term has been intensively investigated by many authors (see [31,32,33]). In particular, [31] deals with existence solutions in the sense distributions of the nonlinear inhomogeneous porous medium type equations
ut−divA(x,t,u,Du)=μinQ:=Ω×(0,T) | (A.3) |
where μ is a nonnegative Radon measure on Q with μ(Q)<∞ and μ|RN+1∖Q=0. In last decade, some authors studied the existence, uniqueness and qualitative properties of the Radon measure-valued solutions to the nonlinear parabolic equations under zero Dirichlet or zero Neumann boundary conditions with bounded Radon measure as initial data (e.g. [1,6,7,9,10,11,12,13,15,25] and references therein). Specially, [6] discuss the existence, uniqueness and the regularity of the Radon measure-valued solutions for a class of nonlinear degenerate parabolic equations
{ut=Δθ(u) in Q,θ(u)=0 on S,u0(x,0)=u0 on Ω, | (A.4) |
where u0∈M+(Ω) and the function θ fulfills the assumptions expressed in [6]. The difference between the abovementioned studies and the problem (P) is the presence of the nonzero-Neumann boundary conditions and the source term which is a solution of the linear inhomogeneous heat equations under Neumann boundary conditions with measure data.
In general, the study of the partial differential equations through numerical methods is investigated by several authors (e.g. [47,48,49,50]). In particular, there are some authors who deal with the computation of the measure-valued solutions of the incompressible or compressible Euler equations (see [47,48]). Mostly, the authors employ the numerical experiment corresponding to initial data of the partial differential equations and prove that the resulting approximation converge to a weak solution. For instance, in [50], the authors study numerical experiment to prove that the convergence of the solution to the nonlinear degenerate parabolic equations is measure-valued. Similarly, [49] employs the numerical method to show that the resulting approximation of a non-coercive elliptic equations with measure data converges to a weak solution. Hence, the numerical experiments represent the straightforward application of the theoretical study of the type of the problem (P).
To address the large-time behavior of the Radon measure-valued solutions of the problem (P), we construct the steady-state problem as a nonlinear strongly degenerate elliptic equations given as follows
{−Δψ(U)+U=u0 in Ω,∂ψ(U)∂η=g(U) on ∂Ω, | (E) |
where u0∈M+(Ω) and the function ψ and g satisfy the hypotheses (I) and (A) respectively.
We consider solutions of the problem (E) as maps from Ω to the cone of nonnegative bounded Radon measure on Ω which satisfies (E) in the following sense: For a suitable class of test function φ, there holds
∫Ω∇ψ(Ur)∇φdx+∫ΩUφdx=∫Ωφdu0(x)+∫∂Ωg(Ur)φdH(x) |
(see Definition 2.6), where Ur∈L1(Ω) denotes the density of the absolutely continuous part of U with respect to the Lebesgue measure.
The nonlinear elliptic equations under Neumann boundary conditions with absorption term and a source term has been intensively studied by several authors [26,34,38,39,40]. In these studies, the authors dealt with the existence, uniqueness and regularity of the solutions. Furthermore, in [34], the following problem is considered
{LU+B(U)=f2 in Ω,∂U∂η+C(U)=g2 on ∂Ω, | (A.5) |
where B(U)∈L1(Ω), C(U)∈L1(∂Ω), f2∈L1(Ω), g2∈L1(∂Ω) and the expression of the differential operator L in [34, Section 2]. The authors proved the existence, uniqueness and regularity of the solutions U∈W1,1(Ω) to the problem (A.5) (see [34, Section 4, Theorem 22 and Corollary 21). The difference between the previous studies mentioned above and (A.5) is that we study the nonlinear strongly degenerate elliptic equations and the solutions obtained are Radon measure-valued. However, the existence, uniqueness, and regularity of the Radon measure-valued solutions of the quasilinear degenerate elliptic equations under zero Dirichlet boundary conditions are discussed in detail [13] by considering the following problem
{−div(A(x,U)∇U)+U(x)=¯μ in Ω,U(x)=0 on ∂Ω, | (A.6) |
where ¯μ∈M(Ω) and A(x,U) satisfies the hypothesis in [13]. In this case, the difference between the problem (E) and (A.4) is a boundary conditions with the assumptions on ψ.
In this paper, we study a class of nonlinear parabolic problems involving a forcing term and initial data is a nonnegative Radon measure. In the recent years, there are different papers that investigate these kind of problems in the setting in which the solution is a Radon measure for positive time. This type of study was done for parabolic and hyperbolic equations. One of the main tool is to search a solution by an approximation of the initial data and then try to pass to the limit in a very weak topology. The innovative part of this work is mainly the study of the large time behavior of the solutions. In my opinion, it is essential to highlight that the explicit examples of equations study in this work have not already been dealt with in literature and the novelties of the techniques that they introduced in the work. Finally, the study of the asymptotic behavior is a novelty.
The main difficulty to study the problem (P) is due to the presence of the forcing term which depends on the property solutions of the inhomogeneous heat equation (H).
The main motivation of this study comes from the desire to deal with parabolic equations in which the forcing term can be either Radon measure or Lp(Q)(1≤p<∞) functions. Whence, the idea to consider the linear inhomogeneous heat equation solution with measure data as a forcing term.
To deal with the existence and the uniqueness of the weak solutions to the problem (P), we use the definition of the Radon measure-valued solutions of the parabolic equations and the natural approximation method. In particular, to show the uniqueness of the problem (P), we will distinguish two cases for the forcing term f, either the function is in L2((0,T),H1(Ω)) or the Radon measure on Q. Notice that when the linear inhomogeneous heat equation (H) does not admit an unique solution, the problem (P) has no unique solution as well.
Furthermore, we prove the necessary and sufficient condition between measure data and capacity in order to deal with the existence of the weak solutions to the problem (P).
To establish the decay estimates of the Radon measure-valued solutions to the problem (P), we construct the suitable function and we use it as a test function in the approximation of the problem (P). Then we easily infer the decay estimates after the use of some measure properties.
To address the large-time behavior of the Radon measure-valued solutions of the problem (P), we first show that the problem (E) has a Radon measure-valued solutions in Ω.
To the best of our knowledge no existing result of decay estimates and large-time behavior of Radon measure-valued solutions obtained as limit of the approximation of the problem (P) are known in the literature. Hence, this interesting case will be discussed in this paper. This paper is organized as follows: In the next section, we state the main results, while in Section 3, we present important preliminaries. In Section 4, we study the existence and uniqueness of the heat equation (H). Finally, we prove the main results in the Sections 5-8.
To study the weak solution of the problem (P), we refer to the following definition.
Definition 2.1. For any u0∈M+(Ω) and μ∈M+(Q), a measure u is called a weak solution of problem (P), if u∈M+(Q) such that
(i) u∈L∞((0,T),M+(Ω)),
(ii) ψ(ur)∈L2((0,T),H1(Ω)),
(iii) g(ur)∈L1(S),
(iv) for every ξ∈C1((0,T),C1(Ω)), ξ(⋅,T)=0 in Ω, u satisfies the identity
∫T0⟨u(⋅,t),ξt(⋅,t)⟩Ωdt+∫T0h(t)⟨f(⋅,t),ξ(⋅,t)⟩Ωdt+⟨u0,ξ(⋅,0)⟩Ω+ |
+∫T0⟨g(ur(⋅,t)),ξ⟩∂Ωdt=∫T0⟨∇ψ(ur),∇ξ⟩Ωdxdt | (2.1) |
where ur is the nonnegative density of the absolutely continuous part of Radon-measure with respect to the Lebesgue measure such that ur∈L∞((0,T),L1(Ω)) and the function f is the solution of the problem (H).
Throughout this paper, we assume that Ω is a strong C1,1 open subset of RN. Also, we assume that there exists a finite open cover (Bj) such that the set Ω∩Bj epigraph of a C1,1 function ζ:RN−1→R that is
Ω∩Bj={x∈Bj/xN>ζ(¯x)}and∂Ω∩Bj={x∈Bj/xN=ζ(¯x)} |
where x=(¯x,xN), the local coordinates with ¯x=(x1,x2,…,xN−1). We denote ϑ={¯x,x∈Ω∩Bj}⊆RN−1, the projection of Ω∩Bj onto the (N−1) first components, and ϑς={¯x,x∈supp(ς)∩Ω}.
If a function ϕ is defined on S, we denote ϕS the function defined on (Bj∩Q)×[0,T] by ξS(x,t)=ξ(¯x,ζ(¯x),t). Notice that the restriction of ξS to [0,T)×ϑ.
The next definition of the trace is corresponding to the problem (P) adapts to the context of [36, Theorem 2.1].
Definition 2.2 Let F∈[L2(Q)]N+1 be such that divF is a bounded Radon measure on Q. Then there exists a linear functional Tη on W12,2(S)∩C(S) which represents the normal traces F⋅ν on S in the sense that the following Gauss-green formula holds:
(i) For all ξ∈C∞c(¯Q),
⟨Tν,ξ⟩=∫QξdivF+∫Q∇ξ⋅F |
where ⟨Tν,ξ⟩ depends only on ξS.
(ii) If (Bj,ς,f) is an above subsequence localization near boundary, then for all ξ∈C∞c([0,T)ׯΩ) there holds
⟨Tν,ξ⟩=−lim | (2.2) |
where the divergence of the fields,
\begin{equation*} \mathcal{F}(x, t) = \left( {\begin{array}{cc} u(x, t) \\ \nabla\psi(u_{r}(x, t)) \\ \end{array} } \right) \end{equation*} |
is a bounded Radon measure on Q .
The following result states the existence of the trace of the boundary condition to the problem (P) .
Lemma 2.1 Let \Omega is a strong C^{1, 1} open subset of \mathbb{R}^{N} . Then there exists an unique trace \mathcal{T}_{\eta}:W^{1, 1}(\Omega)\rightarrow L^{1}(\partial\Omega) such that
\begin{equation*} \langle \mathcal{T}_{\eta}, \xi \rangle = \int_{S}g(u_{r})\xi d\mathcal{H}(x)dt \end{equation*} | (2.3) |
where the function g(u_{r})\in L^{1}(S) and \xi\in C^{\infty}_{c}([0, T)\times\overline{\Omega}) .
To prove the uniqueness of the solution to the problem (P) , we define the notion very weak solution of the problem (P) as follows.
Definition 2.3. For any \mu\in\mathcal{M}^{+}(Q) and u_{0}\in\mathcal{M}^{+}_{d, 2}(\Omega) , a measure u is called a very weak solution to problem (P) if u\in L^{\infty}((0, T), \mathcal{M}^{+}(\Omega)) such that
\begin{equation*} \int_{0}^{T}\langle u(\cdot, t), \xi_{t}(\cdot, t)\rangle_{\Omega}dt+\int_{Q}{\psi(u_{r})\Delta\xi dxdt}+\int_{Q}{h(t)f(x, t)\xi dxdt}+\int_{S}g(u)\xi d\mathcal{H}dt+\langle u_{0}, \xi(0)\rangle_{\Omega} = 0 \end{equation*} | (2.4) |
for every \xi\in C^{2, 1}(\overline{Q}) , which vanishes on \partial\Omega\times[0, T] , for t = T .
To prove the uniqueness of the problem (P) when f lies in \mathcal{M}^{+}(Q) , we consider the following every weak solution gives below:
Definition 2.4 Let u_{0}\in \mathcal{M}_{d, 2}^{+}(\Omega) and \mu\in \mathcal{M}^{+}(Q) such that
\begin{equation*} u_{0} = f_{0}-\text{div}G_{0}\;, \;\, \, f_{0}\in L^{1}(\Omega)\, \, \, \text{and}\, \, \, G_{0}\in \left[L^{2}(\Omega)\right]^{N}. \end{equation*} |
A function u is called a very weak solutions obtained as limit of approximation, if
\begin{equation*} u_{n}\overset{*}\rightharpoonup u\, \, \, \text{in}\, \, \mathcal{M}^{+}(Q) \end{equation*} | (2.5) |
where \left\{u_{n}\right\}\subseteq L^{\infty}(Q)\cap L^{2}((0, T), H^{1}(\Omega)) is the sequences of weak solutions to problem (P_{n}) satisfies
\begin{equation*} \begin{cases} u_{0n} = f_{0n}-F_{0n}\in C^{\infty}_{c}(\Omega), \\ F_{0n}\rightarrow \text{div}G_{0}\quad\text{in}\, \, \, (H^{1}(\Omega))^{*}, \\ f_{0n}\rightarrow f_{0}\quad\text{in}\, \, \, L^{1}(\Omega). \end{cases} \end{equation*} | (2.6) |
We denote (H^{1}(\Omega))^{*} the dual space of H^{1}(\Omega) and the embedding H^{1}(\Omega)\subset L^{2}(\Omega)\subset (H^{1}(\Omega))^{*} holds.
Definition 2.5 Let u_{0}\in \mathcal{M}_{d, 2}^{+}(\Omega) and \mu\in \mathcal{M}_{d, 2}^{+}(Q) such that
\begin{equation*} u_{0} = f_{0}-\text{div}G_{0}\;, \;\, \, f_{0}\in L^{1}(\Omega)\, \, \, \text{and}\, \, \, G_{0}\in \left[L^{2}(\Omega)\right]^{N}. \end{equation*} |
\begin{equation*} \mu = f_{1}-\text{div}G+\varphi_{t}\;, \; f_{1}\in L^{1}(Q)\, \, , \, \, G\in \left[L^{2}(Q)\right]^{N}\quad\text{and}\quad \varphi\in L^{2}((0, T), H^{1}(\Omega)). \end{equation*} |
A measure f is called a very weak solutions obtained as limit of approximation, if
\begin{equation*} f_{n}\overset{*}\rightharpoonup f\, \, \, \text{in}\, \, \mathcal{M}^{+}(Q) \end{equation*} | (2.7) |
where \left\{u_{n}\right\} and \left\{f_{n}\right\}\subseteq L^{\infty}(Q)\cap L^{2}((0, T), H^{1}(\Omega)) are the sequences of weak solutions to problem (P_{n}) and (H_{n}) respectively satisfy
\begin{equation*} \begin{cases} \mu_{n} = f_{1n}-F_{n}+g_{nt}\in C^{\infty}_{c}(Q), \\ u_{0n} = f_{0n}-F_{0n}\in C^{\infty}_{c}(\Omega), \\ f_{1n}\rightarrow f_{1}\quad\text{in}\, \, \, L^{1}(Q), \\ F_{n}\rightarrow \text{div}G\quad\text{in}\, \, \, L^{2}((0, T), (H^{1}(\Omega))^{*}), \\ \varphi_{n}\rightarrow \varphi\quad\text{in}\, \, \, L^{2}((0, T), H^{1}(\Omega)), \\ F_{0n}\rightarrow \text{div}G_{0}\quad\text{in}\, \, \, (H^{1}(\Omega))^{*}, \\ f_{0n}\rightarrow f_{0}\quad\text{in}\, \, \, L^{1}(\Omega). \end{cases} \end{equation*} | (2.8) |
Then, the function u is very weak solutions of the problem (P) obtained as limit of approximation if the function f is a very weak solutions of the problem (H) obtained as limit of approximation.
Notice that
u_{n}\overset{*}\rightharpoonup u\quad\text{in}\, \, \, \mathcal{M}^{+}(Q)\, , \, \, \mu_{n}\overset{*}\rightharpoonup \mu\quad\text{in}\, \, \, \mathcal{M}^{+}(Q)\quad\text{and}\quad u_{0n}\overset{*}\rightharpoonup u_{0}\quad\text{in}\, \, \, \mathcal{M}^{+}(\Omega). |
\mathcal{M}_{d, 2}^{+}(\Omega) denotes the set of nonnegative measures on \Omega which are diffuse with respect to the Newtonian capacity and the definition of the diffuse measure with respect to the parabolic capacity \mathcal{M}_{d, 2}^{+}(Q) will be recalled in the Section 3.
Before dealing with the existence of the problem (P) , we first prove the existence and uniqueness of the solutions to the problem (H) given by the following result.
Theorem 2.1. Assume that u_{0}\in \mathcal{M}^{+}(\Omega) and \mu\in \mathcal{M}^{+}(Q) hold.
(i) Then, there exists a nonnegative Radon measure-valued solution to the problem (H) in the space L^{\infty}((0, T), \mathcal{M}^{+}(\Omega)) such that
\begin{equation*} f(x, t) = \int_{\Omega}G_{N}(x-y, t)du_{0}(y)+ \int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)d\mu(y, \sigma)+ \int_{0}^{t}\int_{\partial\Omega}G_{N}(x-y, t-\sigma)g(f(y, \sigma))d\mathcal{H}(y)d\sigma \end{equation*} | (2.9) |
for almost every t\in(0, T) . Furthermore, the Radon measure-valued solution f satisfies the following estimate
\begin{equation*} \parallel f(\cdot, t) \parallel_{\mathcal{M}^{+}(\Omega)}\leq e^{Ct}\left(\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}+\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}\right) \end{equation*} | (2.10) |
for any C = C(T) a positive constant.
(ii) Suppose that u_{0}\in \mathcal{M}^{+}_{d, 2}(\Omega) , \mu\in \mathcal{M}^{+}_{d, 2}(Q) and g(f) = \overline{K} almost everywhere on S ( \overline{K} is a positive constant) are satisfied. Then, the nonnegative weak Radon measure-valued solution to the problem (H) obtained as limit of the approximation is unique in L^{\infty}((0, T), \mathcal{M}^{+}(\Omega)) .
We denote by G_{N}(x-y, t-s) as the Green function of the heat equation under homogeneous Neumann boundary conditions. By [4], the Green function satisfies the following properties
\begin{equation*} G_{N}(x-y, t-s)\geq0\, , \quad x, y\in\Omega\;, \;0\leq s < t < T, \end{equation*} | (2.11) |
\begin{equation*} \int_{\Omega}G_{N}(x-y, t-s)dx = 1\, , \quad y\in\Omega\;, \;0\leq s < t < T. \end{equation*} | (2.12) |
There exist two positive constants \tau_{1} and \tau_{2} such that
\begin{equation*} \left|G_{N}(x-y, t-s)-\frac{1}{\mid \Omega\mid}\right|\leq \tau_{1}e^{-\tau_{2}(t-s)}\, , \quad x, y\in\Omega\;, \; 1+s < t. \end{equation*} | (2.13) |
\begin{equation*} \lim\limits_{t\rightarrow s}\int_{\Omega}G_{N}(x-y, t-s)\phi(y)dy = \phi(x) \end{equation*} | (2.14) |
for any \phi\in C_{c}(\Omega) and \mid \Omega\mid is a Lebesgue measure of the set \Omega .
Remark 2.1 (i) For any test function \xi\in C^{1}((0, T), C^{1}(\Omega)) such that \xi(\cdot, T) = 0 in \Omega and \frac{\partial\xi}{\partial\eta} = 0 on S , the inner product \langle f(\cdot, t), \xi(\cdot, t) \rangle_{\Omega} in (2.1) is given by the following expression
\begin{equation*} \langle f(\cdot, t), \xi(\cdot, t)\rangle_{\Omega} = \int_{\Omega}\int_{\Omega}G(x-y, t)\xi(y, 0)du_{0}(y)dx+ \end{equation*} |
\begin{equation*} +\int_{\Omega}\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\left(f\xi_{\sigma}-2\nabla f\nabla\xi-f\Delta\xi\right)dyd\sigma dx+ \end{equation*} |
\begin{equation*} +\int_{\Omega}\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\xi(y, \sigma)d\mu(y, \sigma)dx+ \end{equation*} |
\begin{equation*} +\int_{\Omega}\int_{0}^{t}\int_{\partial\Omega}G_{N}(x-y, t-\sigma)\xi(y, \sigma)g(f(y, \sigma))d\mathcal{H}(y)d\sigma dx \end{equation*} | (2.15) |
where \xi_{\sigma} is a first derivative order of \xi with respect \sigma .
(ii) By the regularity properties of the Green function G_{N}(x-y, t-\sigma) in [42], the solution of the problem (H) given by (2.9), f\in L^{2}((0, T), H^{1}(\Omega)) .
(iii) By virtue of the assumptions (J), (2.11) and (2.12), the term h(t)f(x, t) is well-defined at t = 0 . Indeed, the function t\mapsto \int_{\Omega}G_{N}(x-y, t-\sigma)h(\sigma)d\mu(y, \sigma) , t\mapsto \int_{\partial\Omega}G_{N}(x-y, t-\sigma)h(\sigma)g(f(y, \sigma))d\mathcal{H}(y) and t\mapsto \int_{\Omega}G_{N}(x-y, t-\sigma)f(y, \sigma)h'(\sigma)dy are continuous in \mathbb{R}_{+} . Then there holds
\begin{equation*} \lim\limits_{t\rightarrow 0^{+}}h(t)f(x, t) = \lim\limits_{t\rightarrow 0^{+}}\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)f(y, \sigma)h'(\sigma)dyd\sigma+ \end{equation*} |
\begin{equation*} +\lim\limits_{t\rightarrow 0^{+}}\int_{0}^{t}\int_{\partial\Omega}G_{N}(x-y, t-\sigma)h(\sigma)g(f)d\mathcal{H}(y)d\sigma+ \lim\limits_{t\rightarrow 0^{+}}\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)h(\sigma)d\mu(y, \sigma) = 0. \end{equation*} |
Hence we extend the function h(t)f(x, t) in [0, T] defining h(0)f(x, 0) = 0 . Furthermore, the presence of the function h is to well-defined the forcing term of the nonlinear parabolic problem (P) .
In order to study the existence and uniqueness of the solutions to the problem (P) , we give the necessary and sufficient condition on the measures \mu and u_{0} for the existence of the weak solutions to the problem (P) with respect to the parabolic and Newtonian capacity respectively. This result is given by the following theorem.
Theorem 2.2. Suppose that the hypotheses (I) , (A) , \mu\in \mathcal{M}^{+}(Q) and u_{0}\in \mathcal{M}^{+}(\Omega) hold. For any function h satisfying (J), there exists t\in(0, T) such that \int_{0}^{t}h(\sigma)d\sigma = 1 and u is a weak solution to the problem (P) . Then \mu and u_{0} are absolutely continuous measures with respect to the parabolic capacity.
Notice that Newtonian and parabolic capacity are equivalent, then \mu and u_{0} are absolutely continuous measures with respect to C_{2} -capacity as well.
In the next theorem, we present the result of the existence Radon measure-valued solutions to the problem (P) .
Theorem 2.3 Suppose that the assumptions (I) , (J) , (A) \mu\in \mathcal{M}^{+}(Q) and u_{0}\in \mathcal{M}^{+}(\Omega) are satisfied. Then there exists a weak solution u to problem (P) obtained as a limiting point of the sequence \left\{u_{n}\right\} of solutions to problems (P_{n}) such that for every t\in (0, T)\setminus H^{*} , there holds
\begin{equation*} \parallel u(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq C\left(\parallel\mu\parallel_{\mathcal{M}^{+}(Q)}+\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}\right). \end{equation*} | (2.16) |
The result of the uniqueness of the problem (P) is given by the following theorem:
Theorem 2.4 Assume that the hypotheses (I) , (J) and (A) , \mu\in \mathcal{M}^{+}_{d, 2}(Q) and u_{0}\in \mathcal{M}^{+}_{d, 2}(\Omega) hold. Then there exists a unique very weak solution obtained as the limit of approximation u of the problem (P) , if g(u_{r}) = L almost everywhere in S , whenever L is a positive constant.
To establish the decay estimate of the solution to the problem (P) , we recall two particular problems of the problem (P) . Now we consider the following problem.
\begin{equation*} \begin{cases} v_{t} = \Delta\vartheta(v) \ \ &\text{in} \ \ Q, \\ \frac{\partial\vartheta(v)}{\partial\eta} = g_{1}(v) \ \ &\text{on} \ \ S, \\ v(x, 0) = u_{0} \ \ &\text{in} \ \ \Omega, \end{cases} \end{equation*} | (P_{0}) |
The functions \psi and g satisfy the assumption (I) and (A) respectively and have the same properties with the functions \vartheta and g_{1} given as follows
\begin{equation*} \vartheta(s) = \gamma\left[1-\frac{1}{(1+ s)^{m}}\right]\;\;(m > 0)\quad\text{and}\quad g_{1}(s) = \frac{1}{(1+s)^{m}} \end{equation*} | (2.17) |
where m > 0 and s > 0 . Therefore, by Theorem 2.3, the problem (P_{0}) possesses a solution in the space L^{\infty}((0, T), \mathcal{M}^{+}(\Omega)) , such that
\begin{equation*} \parallel v(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq C\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)} \end{equation*} |
for almost every t\in(0, T) .
Similarly, we consider the following problem
\begin{equation*} \begin{cases} w_{t} = \Delta\psi(w)+h(t)f(x, t) \ \ &\text{in} \ \ Q, \\ \frac{\partial\psi(w)}{\partial\eta} = g(w) \ \ &\text{on} \ \ S, \\ w(x, 0) = 0 \ \ &\text{in} \ \ \Omega, \end{cases} \end{equation*} | (P_{1}) |
By Theorem 2.3, the problem (P_{1}) admits a solution in L^{\infty}((0, T), \mathcal{M}^{+}(\Omega)) , such that
\begin{equation*} \parallel w(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq C\parallel \mu \parallel_{\mathcal{M}^{+}(\Omega)} \end{equation*} |
for almost every t\in(0, T) .
Now we state the decay estimates in the next theorem:
Theorem 2.5 Suppose that (I) , (J) , (A) , \mu\in \mathcal{M}^{+}(Q) and u_{0}\in \mathcal{M}^{+}(\Omega) are satisfied. The measure u is the weak solution to the problem (P) . According to Theorem 2.3, v is the weak solution to the problem (P_{0}) and w is the weak solution to the problem (P_{1}) . Then for every t\in (0, T)\setminus H^{*} with \mid H^{*}\mid = 0 , there holds
\begin{equation*} \parallel u(\cdot, t)-v(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq \frac{C}{(T-t)^{\alpha}}\left(\parallel\mu\parallel_{\mathcal{M}^{+}(Q)}+\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}\right), \end{equation*} | (2.18) |
\begin{equation*} \parallel u(\cdot, t)-w(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq C\frac{\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}}{(T-t)^{\alpha}}, \end{equation*} | (2.19) |
and
\begin{equation*} \parallel u(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq \frac{C}{t^{\alpha}}(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)} +\parallel\mu\parallel_{\mathcal{M}^{+}(Q)}) \end{equation*} | (2.20) |
for any positive constant C and \alpha > 1 .
To deal with the large-time behavior of the Radon measure-valued solutions to the problem (P) , we first extend (0, T) to (0, +\infty) , then we assume that the hypothesis
\begin{equation*} \limsup\limits_{t\rightarrow +\infty}\parallel u(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq C \end{equation*} | (2.21) |
where C is a positive constant.
To analyze the large-time behavior of the Radon measure-valued solutions, we first study the existence of the Radon measure-valued solutions corresponding to the steady state problem (E) by considering the following definition.
Definition 2.6 Assume that the hypotheses (I) , (A) and u_{0}\in \mathcal{M}^{+}(\Omega) are satisfied. A measure U is a solution of the problem (E) , if U\in \mathcal{M}^{+}(\Omega) such that
(i) \psi(U_{r})\in W^{1, 1}(\Omega) ,
(ii) g(U_{r})\in L^{1}(\partial\Omega) ,
(iii) for every \varphi\in C^{1}(\Omega) , the following assertion
\begin{equation*} \int_{\Omega}\nabla\psi(U_{r}(x))\nabla \varphi(x)dx+\int_{\Omega}U(x) \varphi(x)dx = \int_{\Omega}\varphi(x)du_{0}(x)+\int_{\partial\Omega}g(U(x))\varphi(x) d\mathcal{H}(x) \end{equation*} | (2.22) |
holds true.
The existence result of the problem (E) is given by the following theorem:
Theorem 2.6 Suppose that the hypotheses (I) , (A) and u_{0}\in \mathcal{M}^{+}(\Omega) are satisfied. Then there exists a weak solution U\in\mathcal{M}^{+}(\Omega) of the problem (E) obtained as a limiting point of the sequence \{U_{n}\} of solutions to the approximation problem (E_{n}) such that
\begin{equation*} \parallel U\parallel_{\mathcal{M}^{+}(\Omega)}\leq C \parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)} \end{equation*} | (2.23) |
where C > 0 is a constant.
The result of the large-time behavior of the Radon measure-valued solutions of the problem (P) is given by the following theorem
Theorem 2.7. Suppose that the assumption (I) , (A) , (J) , u_{0}\in \mathcal{M}^{+}(\Omega) and \mu\in \mathcal{M}^{+}(Q) . U is a Radon measure-valued solutions of the steady-state problem (E) in sense of Theorem 2.6 and u is a Radon measure-valued solutions in the sense of Theorem 2.3 such that (2.21) holds. Then there holds
\begin{equation*} u(\cdot, t)\rightarrow U\quad\text{in}\quad \mathcal{M}^{+}(\Omega) \quad\text{as}\quad t\rightarrow \infty \end{equation*} | (2.24) |
In the following section, we define the truncation function for k > 0 and s\in \mathbb{R} ,
\begin{equation*} T_{k}(s) = \min\{\mid s\mid, k\}\text{sign}(s). \end{equation*} |
To prove the main results from the previous section, we need to recall the preliminaries about capacity and measure collected in [9,10,11,12,13,14,15,16]. Likewise, we recall some important notations as follows:
For any Borel set E\subset\Omega , the C_{2} -capacity of E in \Omega is defined as
C_{2}(E) = \inf\left\{\int_{\Omega}(\mid u\mid^{2}+\mid\nabla u\mid^{2})dx/ u\in \mathbb{Z}_{\Omega}^{E}\right\} |
where \mathbb{Z}_{\Omega}^{E} denotes the set of u which belongs to H^{1}(\Omega) such that 0\leq u\leq 1 almost everywhere in \Omega , and u = 1 almost everywhere in a neighborhood E .
Let W = \left\{u\in L^{2}((0, T), H^{1}(\Omega))\; \; \; \text{and}\; \; u_{t}\in L^{2}((0, T), (H^{1}(\Omega))^{*})\right\} endowed with its natural norm \parallel u\parallel_{W} = \parallel u\parallel_{L^{2}((0, T), H^{1}(\Omega))}+\parallel u_{t}\parallel_{L^{2}((0, T), (H^{1}(\Omega))^{*})} a Banach space. For any open set U\subset Q , we define the parabolic capacity as
\text{Cap}(U) = \inf\left\{\parallel u\parallel_{W}/ u\in \mathbb{V}_{Q}^{U}\right\} |
where \mathbb{V}_{Q}^{U} denotes the set of u belongs to W such that 0\leq u\leq 1 almost everywhere in Q , and u = 1 almost everywhere in a neighborhood U .
Let \mathcal{M}(B) be the space of bounded Radon measures on B , and \mathcal{M}^{+}(B)\subset \mathcal{M}(B) the cone of nonnegative bounded Radon measures on B . For any \mu\in\mathcal{M}(B) a bounded Radon measure on B , we set
\parallel \mu\parallel_{\mathcal{M}(\Omega)}: = \mid \mu\mid(B) |
where \mid \mu\mid stands for the total variation of \mu .
The duality map \langle\cdot, \cdot\rangle_{B} between the space \mathcal{M}(B) and C_{c}(B) is defined by
\langle\mu, \varphi \rangle_{B} = \int_{B}\varphi d\mu. |
\mathcal{M}_{s}^{+}(B) denotes the set of nonnegative measures singular with respect to the Lebesgue measure, namely
\mathcal{M}_{s}^{+}(B): = \left\{\mu\in \mathcal{M}^{+}(\Omega)/ \exists\text{ a Borel set}\; F\subseteq B\;\text{such that}\; \mid F\mid = 0\;, \mu = \mu\llcorner B\right\} |
we will consider either \mid.\mid the Lebesgue measure on \mathbb{R}^{N} or \mathbb{R}^{N+1} . Similarly, \mathcal{M}_{ac}^{+}(B) the set of nonnegative measures absolutely continuous with respect to the Lebesgue measure, namely
\mathcal{M}_{ac}^{+}(B): = \left\{\mu\in \mathcal{M}^{+}(\Omega)/\mu(F) = 0, \text{ for every Borel set}\, F\subseteq B\, \text{such that}\, \mid F\mid = 0\right\} |
Let \mathcal{M}_{c, 2}^{+}(B) be the set of nonnegative measures on B which are concentrated with respect to the Newtonian capacity
\mathcal{M}_{c, 2}^{+}(B): = \left\{\mu\in \mathcal{M}^{+}(B)/\exists\text{ a Borel set}\; F\subseteq B, \;\text{such that}\;\mu = \mu\llcorner F\; \text{and}\; C(F) = 0\right\} |
\mathcal{M}_{d, 2}^{+}(B) denotes the set of nonnegative measures on B which are diffuse with respect to the Newtonian capacity
\mathcal{M}_{d, 2}^{+}(B): = \left\{\mu\in \mathcal{M}^{+}(B)/\mu(F) = 0, \text{ for every Borel set}\, F\subseteq B\;\text{such that}\; C(F) = 0\right\}. |
It is known that a measure \overline{\mu}_{d, 2}\in \mathcal{M}_{d, 2}^{+}(\Omega) (resp. \mu_{d, 2}\in \mathcal{M}_{d, 2}^{+}(Q) ) if there exist f_{0}\in L^{1}(\Omega) and G_{0}\in \left[L^{2}(B)\right]^{N} (resp. if \mu_{d, 2}\in \mathcal{M}_{d, 2}^{+}(Q) , there exist f\in L^{1}(Q) , g\in L^{2}((0, T), H^{1}(\Omega)) and G\in \left[L^{2}(Q)\right]^{N} ) such that
\begin{equation*} \overline{\mu}_{d, 2} = f_{0}-\text{div}G_{0}\;\; \text{in}\quad D'(\Omega)\;\;(\text{resp.}\; \mu_{d, 2} = f-\text{div}G+g_{t}\;\; \text{in}\quad D'(Q)). \end{equation*} | (3.1) |
For any \lambda\in \mathcal{M}^{+}(B) , if there exists a unique couple \lambda_{d, 2}\in \mathcal{M}_{d, 2}^{+}(B) , \lambda_{c, 2}\in \mathcal{M}_{c, 2}^{+}(B) such that
\begin{equation*} \lambda = \lambda_{d, 2}+\lambda_{c, 2}. \end{equation*} | (3.2) |
On the other hand, there exists a unique couple \; \lambda_{ac}\in \mathcal{M}_{ac}^{+}(B), \; \lambda_{s}\in \mathcal{M}_{s}^{+}(B) such that
\begin{equation*} \lambda = \lambda_{ac}+\lambda_{s} \end{equation*} | (3.3) |
where either B = \Omega , C(F) = C_{2}(E) or B = Q , C(F) = Cap(U) .
By L^{\infty}\left((0, T), \mathcal{M}^{+}(\Omega)\right) , the set of nonnegative Radon measures u\in \mathcal{M}^{+}(\overline{Q}) such that for every t\in (0, T) , there exists a measure u(\cdot, t)\in \mathcal{M}^{+}(\Omega) such that
(i) for every \xi\in C(\overline{Q}) the map
\begin{equation*} t\mapsto\langle u(\cdot, t), \xi(\cdot, t) \rangle_{\Omega}\;\text{is Lebesgue measurable} \end{equation*} |
and
\begin{equation*} \langle u(\cdot, t), \xi(\cdot, t) \rangle_{\Omega} = \int_{0}^{T}\langle u(\cdot, t), \xi(\cdot, t) \rangle_{\Omega}dt \end{equation*} |
(ii) there exists a constant C > 0 such that
\begin{equation*} \text{ess}\sup\limits_{t\in(0, T)}\parallel u(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq C \end{equation*} |
with the norm denotes
\begin{equation*} \parallel u\parallel_{L^{\infty}((0, T), \mathcal{M}^{+}(\Omega))} = \text{ess}\sup\limits_{t\in(0, T)}\parallel u(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}. \end{equation*} | (3.4) |
In the literature, many authors dealt with the existence, uniqueness, blow-up at finite and infinite time, decay estimates, stability properties and asymptotic behavior of the solutions to the heat equation under Neumann boundary conditions with a source term and initial data, such as (see [2,3,4,5,42,43] and references therein). Moreover, most of the authors employed the maximum principle theorem through the monotonicity technique and semi-group method to show the existence, blow-up, stability properties and asymptotic behavior of these solutions. Meanwhile, in this section we prove the existence and uniqueness of the linear inhomogeneous heat equation (H) by using the fundamental solution of the heat equation (see [2,3,4,42]). Also, we use the definition of the Radon measure-valued solutions in [9] and some properties of the Radon measure provided in [24,44]. Moreover, we consider for every n\in \mathbb{N} , the approximation of problem (H) such that
\begin{equation*} \begin{cases} f_{nt} = \Delta f_{n}+\mu_{n} \ \ &\text{in} \ \ Q, \\ \frac{\partial f_{n}}{\partial\eta} = g(f_{n}) \ \ &\text{on} \ \ S, \\ f_{n}(x, 0) = u_{0n}(x) \ \ &\text{in} \ \ \Omega, \end{cases} \end{equation*} | (H_{n}) |
Since u_{0}\in \mathcal{M}^{+}(\Omega) , the approximation of the Radon measure u_{0} is given by [9, Lemma 4.1] such that \{u_{0n}\}\subseteq C_{c}^{\infty}(\Omega) satisfies the following assumptions
\begin{equation*} \begin{cases} u_{0n}\overset{*}\rightharpoonup u_{0}\quad\text{in}\quad \mathcal{M}^{+}\Omega), \\ u_{0n}\rightarrow u_{0r}\quad\text{a.e in}\quad\Omega, \\ \parallel u_{0n}\parallel_{L^{1}(\Omega)}\leq \parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}. \end{cases} \end{equation*} | (4.1) |
Moreover \mu\in \mathcal{M}^{+}(Q) , the approximation of the Radon measure \mu is given by [15] such that \{\mu_{n}\}\subseteq C_{c}^{\infty}(Q) fulfills the following hypotheses
\begin{equation*} \begin{cases} \mu_{n}\overset{*}\rightharpoonup \mu\quad\text{in}\quad \mathcal{M}^{+}(Q), \\ \mu_{n}\rightarrow \mu_{r}\quad\text{a.e in}\quad Q, \\ \parallel \mu_{n}\parallel_{L^{1}(Q)}\leq \parallel \mu\parallel_{\mathcal{M}^{+}(Q)}, \end{cases} \end{equation*} | (4.2) |
for every n\in \mathbb{N} . By [21,22,43], the approximation problem (H_{n}) has a unique solution f_{n} in C^{1}((0, T), L^{2}(\Omega))\cap L^{2}((0, T), H^{1}(\Omega))\cap L^{\infty}(Q) .
In the next proposition, we establish the relationship between the approximation solution f_{n} and any test function in (P_{n}) .
Proposition 4.1. Suppose that \xi\in C^{1}_{c}(Q) such that \frac{\partial\xi}{\partial\eta} = 0 on S , the test function in (H_{n}) and f_{n} the approximation solution of the problem (H_{n}) . Then, the following expression holds
\begin{equation*} f_{n}(x, t)\xi(x, t) = \int_{\Omega}G_{N}\left(x-y, t\right)\xi(y, 0)u_{0n}(y)dy+ \int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\left\{ f_{n}\xi_{\sigma}-2\nabla f_{n}\nabla\xi-f_{n}\Delta\xi\right\}dyd\sigma+ \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\xi(y, \sigma)\mu_{n}(y, \sigma)dyd\sigma+ \int_{0}^{t}\int_{\partial\Omega}G_{N}(x-y, t-\sigma)\xi(y, \sigma)g(f(y, \sigma))d\mathcal{H}(y)d\sigma \end{equation*} | (4.3) |
where \xi_{\sigma} is first-order derivative order of \xi with respect to \sigma .
Remark 4.2. Assume that the test function \xi = \rho\in C^{2}_{c}(\Omega) , then we obtain
\begin{equation*} f_{n}(x, t)\rho(x) = \int_{\Omega}G_{N}\left(x-y, t\right)\rho(y)u_{0n}(y)dy- \int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\left\{ 2\nabla f_{n}\nabla\rho+f_{n}\Delta\rho\right\}dyd\sigma+ \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\rho(y)\mu_{n}(y, \sigma)dyd\sigma+ \int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\rho(y)g(f(y, \sigma))d\mathcal{H}(y)d\sigma. \end{equation*} | (4.4) |
On the other hand, we suppose that the test function \xi = \theta(t)\in C^{1}(0, T) then (4.3) reads
\begin{equation*} f_{n}(x, t)\theta(t) = \int_{\Omega}G_{N}\left(x-y, t\right)\theta(0)u_{0n}(y)dy+ \int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)f_{n}(y, \sigma)\theta'(\sigma)dyd\sigma+ \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\theta(\sigma)\mu_{n}(y, \sigma)dyd\sigma+ \int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\theta(\sigma)g(f(y, \sigma))d\mathcal{H}(y)d\sigma. \end{equation*} | (4.5) |
Proof of Proposition 4.1. Assume that \xi\in C^{1}((0, T), C^{2}_{c}(\Omega)) such that \frac{\partial\xi}{\partial\eta} = 0 on S , a test function in (H_{n}) , then the following equation
\begin{equation*} \begin{cases} (f_{n}\xi)_{t} = \Delta(f_{n}\xi)+ f_{n}\xi_{t}-2\nabla f_{n}\nabla\xi-f_{n}\Delta\xi+\mu_{n}\xi \ \ &\text{in} \ \ \Omega\times(0, T), \\ \frac{\partial(f_{n}\xi)}{\partial\eta} = g(f_{n})\xi \ \ &\text{on} \ \ \partial\Omega\times(0, T), \\ f_{n}(x, 0)\xi(x, 0) = u_{0n}\xi(x, 0) \ \ &\text{in} \ \ \Omega, \end{cases} \end{equation*} | (H_{\xi}) |
is well-defined. By [35, Chapter 20, Section 20.2], the problem (H_{\xi}) admits a unique solution f_{n}\xi expressed in (4.3).
Proof of Theorem 2.1 (i) We argue this proof into two steps:
Step 1. We show that \{f_{n}(\cdot, t)\} is a Cauchy sequence in L^{1}(\Omega) a.e in (0, T) . To attain this, we use the expression (4.3) to prove the Cauchy sequence. Indeed, for any m, n\in \mathbb{N} there holds
\begin{equation*} f_{n}(x, t)-f_{m}(x, t) = \int_{\Omega}G_{N}(x-y, t)[u_{0n}(y)-u_{0m}(y)]dy+ \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{\Omega}G_{N}(x-\xi, t-\sigma)[\mu_{n}(\xi, \sigma)-\mu_{m}(y, \sigma))]dyds+ \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{\partial\Omega}G_{N}(x-y, t-\sigma)[g(f_{n}(\xi, \sigma))-g(f_{m}(\xi, s))] d\mathcal{H}(y)ds. \end{equation*} | (4.6) |
From the assumption (2.12), the Eq (4.6) yields
\begin{equation*} \int_{\Omega}\mid f_{n}(x, t)-f_{m}(x, t)\mid dx\leq\int_{\Omega}\mid u_{0n}(y)-u_{0m}(y)\mid dy+ \int_{0}^{t}\int_{\Omega}\mid\mu_{n}(y)-\mu_{m}(y)\mid dyd\sigma \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{\partial\Omega}G_{N}(x-\xi, t-s)d\mathcal{H}(\xi) \left(\int_{\Omega}\mid g(f_{n}(x, s))-g(f_{m}(x, s))\mid dx\right)ds. \end{equation*} | (4.7) |
Furthermore, by using the mean value theorem, we find that there exists a function \theta(x, s) which is continuous in \overline{Q}_{T_{1}} such that \alpha_{1} < \theta(x, s) < \alpha_{2} , g(f_{n}(x, s))-g(f_{m}(x, s)) = g'(\theta(x, s))(f_{n}(x, s)-f_{m}(x, s)) , where g'(\theta(x, s))\in L^{\infty}(\mathbb{R}_{+}) (see assumption (I) -(i)) and 0 < \alpha_{1} < \alpha_{2} are constants, therefore we obtain
\begin{equation*} \int_{\Omega}\mid f_{n}(x, t)-f_{m}(x, t)\mid dx\leq\int_{\Omega}\mid u_{0n}(y)-u_{0m}(y)\mid dy+\int_{0}^{T}\int_{\Omega}\mid\mu_{n}(y)-\mu_{m}(y)\mid dy+ \end{equation*} |
\begin{equation*} +C(T_{1})\int_{0}^{t}\int_{\Omega}\mid f_{n}(x, \sigma)-f_{m}(x, \sigma)\mid dx d\sigma, \end{equation*} | (4.8) |
whenever C(T_{1}) = \sup_{(\xi, \sigma)\in \overline{Q}_{T_{1}}}\int_{\partial\Omega}G_{N}(x-\xi, T_{1})g'(\theta(x, s))d\mathcal{H}(\xi) > 0 . By the property (2.13) of the Green function G_{N}(x-\xi, t-\sigma) of the heat equation with nonhomogeneous Neumann boundary and the fact that g'(\theta(x, s))\in L^{\infty}(\mathbb{R}_{+}) , then C(T_{1}) is a constant depending on T_{1} . From the Gronwall's inquality, the inequality (4.8) yields
\begin{equation*} \int_{\Omega}\mid f_{n}(x, t)-f_{m}(x, t)\mid dx\leq C(T, T_{1})\int_{\Omega}\mid u_{0n}(y)-u_{0m}(y)\mid dy+ \end{equation*} |
\begin{equation*} + C(T, T_{1})\int_{\Omega}\mid\mu_{n}(y, \sigma)-\mu_{m}(y, \sigma)\mid dyd\sigma \end{equation*} | (4.9) |
for a.e 0\leq t < T_{1} < T and C(T, T_{1}) = 1+C(T_{1})Te^{C(T_{1})T} > 0 a constant.
Since the sequences \{u_{0n}\} and \{u_{0m}\} are satisfying the assumption (4.1) and \{\mu_{n}\} and \{\mu_{m}\} are verifying the assumption (4.2), then by passing to the limit as n and m go to infinity, there holds
\begin{equation*} \lim\limits_{n, m\rightarrow +\infty}\int_{\Omega}\mid f_{n}(x, t)-f_{m}(x, t)\mid dx\leq C(T, T_{1})\limsup\limits_{n\rightarrow +\infty}\int_{\Omega}\mid u_{0n}(y)-u_{0r}(y)\mid dy+ \end{equation*} |
\begin{equation*} +C(T, T_{1})\limsup\limits_{m\rightarrow +\infty}\int_{\Omega}\mid u_{0m}(y)-u_{0r}(y)\mid dy+ \end{equation*} |
\begin{equation*} +C(T, T_{1})\limsup\limits_{n\rightarrow +\infty}\int_{\Omega}\mid\mu_{n}(y, \sigma)-\mu_{r}(y, \sigma)\mid dyd\sigma+ \end{equation*} |
\begin{equation*} +C(T, T_{1})\limsup\limits_{m\rightarrow +\infty}\int_{\Omega}\mid\mu_{m}(y, \sigma)-\mu_{r}(y, \sigma)\mid dyd\sigma\leq0. \end{equation*} | (4.10) |
Hence the sequence \{f_{n}(\cdot, t)\} is Cauchy in L^{1}(\Omega) for almost every t\in(0, T) .
Step 2. We show that f_{n}(\cdot, t)\overset{*}\rightharpoonup f(\cdot, t) in \mathcal{M}^{+}(\Omega) a.e in (0, T) .
Since the function f_{n}(x, t) is a solution of the approximation problem (H_{n}) and \mu_{n}(x)\geq 0 in Q , u_{0n}(x)\geq 0 in \Omega , g > 0 in \mathbb{R}_{+} , then we apply the maximum principal theorem in [22,43] and then the solution of the approximation problem (H_{n}) is nonnegative in \overline{Q} . Likewise, we assume that \xi(x, t)\equiv 1 , then we obtain
\begin{equation*} f_{n}(x, t) = \int_{\Omega}G_{N}(x-y, t-\sigma)u_{0n}(y)dy+\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\mu_{n}(y, \sigma)dyd\sigma+ \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{\partial\Omega}G_{N}(x-y, t-\sigma)g(f(y, \sigma))d\mathcal{H}(y)d\sigma. \end{equation*} | (4.11) |
By the assumptions (A) , (2.12), (2.13), (4.1) and (4.2), we infer that
\begin{equation*} \int_{\Omega} f_{n}(x, t)dx\leq \parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(Q)} +C\int_{0}^{t}\int_{\Omega}f_{n}(x, \sigma)dxd\sigma. \end{equation*} | (4.12) |
By Gronwall's inequality, we deduce that
\begin{equation*} \parallel f_{n}(\cdot, t)\parallel_{L^{1}(\Omega)}\leq e^{Ct}\left(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}\right), \end{equation*} | (4.13) |
for almost every t\in(0, T) .
By Step 1, the sequence \{f_{n}(\cdot, t)\} is Cauchy in L^{1}(\Omega) , then we infer that f_{n}(\cdot, t)\rightarrow f(\cdot, t) a.e in (0, T) . Hereby we argue as in [9, Proposition 5.3], one proves that f(\cdot, t)\in\mathcal{M}^{+}(\Omega) and the following convergence
\begin{equation*} f_{n}(\cdot, t)\overset{*}\rightharpoonup f(\cdot, t)\quad\text{in}\quad\mathcal{M}^{+}(\Omega) \end{equation*} | (4.14) |
for almost every t\in(0, T) holds.
From [44, Chapter 5, Section 5.2.1, Theorem 1], the estimate (4.13) yields
\begin{equation*} \parallel f(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq\liminf\limits_{n\rightarrow +\infty}\int_{\Omega} f_{n}(\cdot, t)dx\leq e^{Ct}\left(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}\right). \end{equation*} |
The estimate (2.10) is achieved.
(ii) Now we show the uniqueness solutions to the problem (H).
To attain this, we consider f_{1} and f_{2} two every weak solutions of the problem (H) in sense of Definition 2.5 with initial data u_{01} and u_{02} respectively.
Let \{f_{1n}\}, \; \{f_{2n}\}\subseteq L^{\infty}(Q)\cap L^{2}((0, T), H^{1}(\Omega)) be two weak solutions given by the proof (i) of Theorem 2.1. Assume that \{u_{01n}\}, \; \{u_{02n}\}\; \{\mu_{1n}\}, \; \{\mu_{2n}\} are approximating Radon measures in sense of Definition 2.4 and f_{1n} , f_{2n} in (4.11) hold. Since we have assumed that g(f) = \overline{K} almost everywhere in S , thus g(f_{1n}) = g(f_{2n}) = \overline{K} on S . For any \xi\in C^{1}_{c}(Q) such that \frac{\partial\xi}{\partial\eta} = 0 on S , there holds
\begin{equation*} \int_{0}^{T}[f_{1n}(y, t)-f_{2n}(y, t)]\xi(y, t) dt = \int_{Q}[f_{1n}(x, t)-f_{2n}(x, t)]\delta_{y}(x)\xi(x, t)dxdt \end{equation*} |
\begin{equation*} = \int_{0}^{T}\int_{\Omega}G_{N}\left(0, t\right)(u_{01n}(y)-u_{02n}(y))\xi(y, t)dydt+ \end{equation*} |
\begin{equation*} +\int_{0}^{T}\int_{0}^{t}\int_{\Omega}G_{N}(0, t-\sigma)(\mu_{1n}(y, \sigma)-\mu_{2n}(y, \sigma)) \xi(y, t)dyd\sigma = : I_{1}+I_{2}. \end{equation*} | (4.15) |
Let us evaluate the limit of I_{1} and I_{2} when n\rightarrow \infty . To attain this, we begin with the expression I_{1} :
\begin{equation*} I_{1} = \left[\int_{\Omega}(u_{01n}(y)-u_{01n}(y))\xi(y, t)dy\right] \left[\int_{0}^{T}G_{N}\left(0, t\right)dt\right]. \end{equation*} |
Taking \xi(y, t) = \rho(y)\widetilde{h}(t) with \rho\in C^{2}(\Omega) such that \frac{\partial \rho}{\partial\eta} = 0 on \partial\Omega and \widetilde{h}\in C_{c}(0, T) , then we have
\begin{equation*} I_{1} = \int_{0}^{T}\widetilde{h}(t)G_{N}\left(0, t\right)dt\int_{\Omega}(u_{01n}(y)-u_{02n}(y))\rho(y)dy \end{equation*} |
\begin{equation*} = \int_{0}^{T}\widetilde{h}(t)G\left(0, t\right)dt \left[\int_{\Omega}(f_{01n}(y)-f_{02n}(y))\rho(y)dy -\int_{\Omega}(F_{01n}(y)-F_{02n}(y))\rho(y)dy\right]. \end{equation*} |
Passing to the limit when n\rightarrow \infty , there holds
\begin{equation} \lim\limits_{n\rightarrow \infty}I_{1} = 0. \end{equation} | (4.16) |
Now we consider the expression I_{2} ,
\begin{equation*} I_{2} = \left[\int_{0}^{T}\int_{\Omega}(\mu_{1n}(y, t)-\mu_{2n}(y, t))\xi(y, t)dydt\right] \left[\int_{0}^{t}G(0, t-\sigma)d\sigma\right]. \end{equation*} |
According to Definition 2.4, it is worth observing that
\begin{equation*} I_{2} = \left[\int_{0}^{t}G(0, t-\sigma)d\sigma\right]\left[\int_{Q}(f_{11n}(y, t)-f_{12n}(y, t))\xi(y, t)dydt\right]- \end{equation*} |
\begin{equation*} \quad-\left[\int_{0}^{t}G(0, t-\sigma)d\sigma\right]\left[\int_{Q}(F_{1n}(y, t)-F_{2n}(y, t))\xi(y, t)dydt\right]+ \end{equation*} |
\begin{equation*} \quad+\left[\int_{0}^{t}G(0, t-\sigma)d\sigma\right]\left[\int_{Q}(\varphi_{1n}(y, t)-\varphi_{2n}(y, t))\xi_{t}(y, t)dydt\right]+ \end{equation*} |
\begin{equation*} \quad+\left[\int_{0}^{t}G(0, t-\sigma)d\sigma\right] \left[\int_{\Omega}(\varphi_{1n}(y, 0)-\varphi_{2n}(y, 0))\xi(y, 0)dy\right]. \end{equation*} |
We pass to the limit when n goes to infinity, therefore
\begin{equation} \lim\limits_{n\rightarrow \infty}I_{2} = 0. \end{equation} | (4.17) |
By (4.16), (4.17) and Dominated Convergence theorem, we obtain
\begin{equation} \int_{Q}(f_{1}(x, t)-f_{2}(x, t))\xi(x, t) dxdt = 0, \end{equation} | (4.18) |
which leads to
\begin{equation*} f_{1n}\overset{*}\rightharpoonup f_{1}\;\;\mathcal{M}^{+}(Q)\;\; \text{and}\;\;f_{2n}\overset{*}\rightharpoonup f_{2}\;\;\mathcal{M}^{+}(Q). \end{equation*} |
Hence f_{1} = f_{2} holds.
Remark 4.1 (i) Since f\in \mathcal{M}^{+}(Q) , then it is worthy observing that f_{n} in (4.11) is a sequence of the approximation Radon measure f satisfying the following properties
\begin{equation*} \begin{cases} f_{n}\overset{*}\rightharpoonup f\quad\text{in}\quad \mathcal{M}^{+}(Q), \\ f_{n}\rightarrow f\quad\text{a.e in}\quad Q, \\ \parallel f_{n}\parallel_{L^{1}(Q)}\leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel\mu\parallel_{\mathcal{M}^{+}(Q)}), \\ f\;\; \text{is given in}\;\; (2.9), \;\; \end{cases} \end{equation*} | (4.19) |
for every n\in \mathbb{N} and C > 0 is a constant.
(ii) By (2.11)-(2.13) and the assumption (A), we deduce from the compactness theorem in [23] the approximation problem (H_{n}) possesses a weak solution f in L^{2}((0, T), H^{1}(\Omega)) such that the properties
\begin{equation*} \begin{cases} f_{n} = T_{n}(f), \\ f_{n}\rightarrow f\quad\text{a.e in}\quad Q, \\ \mid f_{n}\mid\leq \mid f\mid, \end{cases} \end{equation*} | (4.20) |
hold true.
Proof of Lemma 2.1. To prove this result, we use Definition 2.2 and we recall the Gauss-green formula given by the functional
\begin{equation*} \langle \mathcal{T}_{\nu}, \xi \rangle = \int_{Q}\xi\text{div}\mathcal{F}+\int_{Q}\nabla\xi\cdot\mathcal{F} \end{equation*} | (5.1) |
Since there exists a linear continuous functional \mathcal{T}_{\nu} on W^{\frac{1}{2}, 2}(S)\cap C(S) which stands for \mathcal{F}\cdot\nu , then we define a notion of the normal trace of the flux \nabla\psi(u_{r})\cdot\nu such that
\begin{equation*} \langle \mathcal{T}_{\eta}, \xi \rangle = \langle \mathcal{T}_{\nu}, \xi \rangle+\int_{\Omega}\xi(x, 0)du_{0}(x)+\int_{Q}h(t)f(x, t)\xi dxdt. \end{equation*} | (5.2) |
The definition make sense because of the definition of the weak solution when we assume that the value of the initial data
\begin{equation*} \lim\limits_{s\rightarrow0}\frac{1}{s}\int_{0}^{s}\int_{\Omega}u(x, t)\xi(x, t)dxdt = \int_{\Omega}\xi(x, 0)du_{0} \end{equation*} | (5.3) |
holds, for any \frac{s-t}{s}\chi_{(0, s)}(t)\phi as a test function in (2.1). In particular \langle \mathcal{T}_{\nu}, \xi\sigma \rangle depends only on \xi_{S} and from (2.2), we infer the formula
\begin{equation*} \langle \mathcal{T}_{\nu}, \xi \rangle = -\lim\limits_{s\rightarrow0}\frac{1}{s}\int_{s}^{T}\int_{\vartheta} \int_{\zeta(\overline{x})}^{\zeta(\overline{x})+s}\nabla\psi(u_{r}) \left( {\begin{array}{cc} -\nabla \zeta(\overline{x}) \\ 1 \\ \end{array} } \right)\xi\varsigma dx_{N}d\overline{x}dt \end{equation*} | (5.4) |
for any \xi\in C^{\infty}_{c}([0, T)\times\overline{\Omega}) . We denote \{v_{\delta}\} a boundary-layer sequence of C^{2}(\Omega)\cap C(\overline{\Omega}) such that
\begin{equation*} \lim\limits_{\delta\rightarrow0^{+}}v_{\delta} = 1\;\;\text{a.e in }\;\;\Omega, \quad 0\leq v_{\delta}\leq 1, \;\; v_{\delta} = 0\quad\text{on}\;\;\partial\Omega. \end{equation*} | (5.5) |
For more properties concerning the boundary-layer sequence \{v_{\delta}\} (see [37, Lemma 5.5 and Lemma 5.7]). If \xi\in\left(H^{1}(\Omega)\right)^{N} , then
\begin{equation*} \lim\limits_{\delta\rightarrow0^{+}}\int_{\Omega}\varsigma\xi\nabla v_{\delta} = -\lim\limits_{\delta\rightarrow0^{+}}\int_{\Omega}\text{div}(\varsigma\xi)v_{\delta} = -\int_{\Omega}\text{div}(\varsigma\xi) = -\int_{\partial\Omega}\varsigma\xi\cdot\eta d\mathcal{H}(x) \end{equation*} | (5.6) |
The previous statement (5.6) explains that for any function-valued F:\Omega\rightarrow \mathbb{R}^{N} , then -F\cdot\nabla v_{\delta} approaches the normal trace F\cdot\eta . Let \xi\in C^{\infty}_{c}([0, T)\times\overline{\Omega}) and \xi = \xi(1-v_{\delta}) on S , it implies that \langle \mathcal{T}_{\nu}, \xi\varsigma \rangle = \langle \mathcal{T}_{\nu}, \varsigma\xi(1-v_{\delta}) \rangle . By Definition 2.2 and the equation (5.2), the Gauss-Green formula yields
\begin{equation*} \langle \mathcal{T}_{\eta}, \varsigma\xi \rangle = \langle \mathcal{T}_{\nu}, \varsigma\xi(1-v_{\delta}) \rangle+\int_{\Omega}\xi(x, 0)\varsigma(1-v_{\delta})du_{0}+ \int_{Q}h(t)f(x, t)\xi(x, t)\varsigma(1-v_{\delta})dxdt \end{equation*} |
\begin{equation*} = \int_{Q}\xi\varsigma(1-v_{\delta})\text{div}\mathcal{F}+ \int_{Q}\nabla(\xi\varsigma(1-v_{\delta}))\cdot\mathcal{F}+\int_{\Omega}\xi(x, 0)\varsigma(1-v_{\delta})du_{0}+ \end{equation*} |
\begin{equation*} +\int_{Q}h(t)f(x, t)\xi(x, t)\varsigma(1-v_{\delta})dxdt. \end{equation*} |
Since 0\leq v_{\delta}\leq 1 and v_{\delta}\rightarrow 1 a.e in \Omega as \delta\rightarrow0^{+} , then Dominated Convergence Theorem ensures that
\begin{equation*} \lim\limits_{\delta\rightarrow0^{+}}\int_{Q}\xi\varsigma(1-v_{\delta})\text{div}\mathcal{F} = 0\quad\text{and}\quad \langle \mathcal{T}_{\eta}, \xi\varsigma \rangle = -\lim\limits_{\delta\rightarrow0^{+}} \int_{Q}\nabla\xi(u_{r})\nabla v_{\delta}\xi\varsigma dxdt \end{equation*} | (5.7) |
On the other hand, we consider \xi\varsigma(1-v_{\delta}) as a test function in the problem (P) then the following expression holds
\begin{equation*} -\int_{\Omega}\phi(x, 0)\varsigma(1-v_{\delta})du_{0}+\int_{\Omega}u(x, T)\xi(x, T)\varsigma(1-v_{\delta})dx -\int_{Q}u(x, t)\xi_{t}(x, t)\varsigma(1-v_{\delta})dxdt \end{equation*} |
\begin{equation*} = -\int_{Q}\nabla\psi(u_{r})\nabla(\xi\varsigma)(1-v_{\delta})dxdt+ \int_{Q}\nabla\psi(u_{r})\nabla v_{\delta}\xi\varsigma dxdt+\int_{S}g(u_{r})\xi\varsigma d\mathcal{H}(x)dt+. \end{equation*} |
\begin{equation*} +\int_{Q}h(t)f(x, t)\xi(x, t)\varsigma(1-v_{\delta})dxdt. \end{equation*} |
Since 0\leq v_{\delta}\leq 1 and v_{\delta}\rightarrow 1 a.e in \Omega as \delta\rightarrow0^{+} , then Dominated Convergence Theorem yields
\begin{equation*} \int_{S}g(u_{r})\xi\varsigma d\mathcal{H}(x)dt = -\lim\limits_{\delta\rightarrow0^{+}}\int_{Q}\nabla\psi(u_{r})\nabla v_{\delta}\xi\varsigma dxdt. \end{equation*} | (5.8) |
By combining the assertions (5.7) with (5.8), the statement (2.3) is satisfied.
Proof of Theorem 2.2. Assume that for any compact set K = K_{0}\times[0, T]\subset\Omega\times(0, T)\subset \mathbb{R}^{N}\times \mathbb{R}_{+} (resp. for any compact set K_{0}\subset\Omega\subset \mathbb{R}^{N} ) such that \mu^{-}(K) = 0 (resp. u_{0}^{-}(K_{0}) = 0) and Cap (K) = 0 (resp. C_{2}(K_{0}) = 0 ). To show that \mu and u_{0} are absolutely continuous measures with respect to the parabolic capacity, it is enough to prove that \mu^{+}(K) = 0 (resp. u^{+}_{0}(K_{0}) = 0) . To this purpose Cap (K) = 0 (resp. C_{2}(K_{0}) = 0 ), there exists a sequence \{\varphi_{n}(t)\}\subset C^{\infty}_{c}(\mathbb{R}^{N}\times \mathbb{R}_{+}) (resp. \{\varphi_{n}(0)\}\subset C^{\infty}_{c}(\mathbb{R}^{N} )) such that 0\leq \varphi_{n}(t)\leq 1 in Q (resp. 0\leq \varphi_{n}(0)\leq 1 in \Omega ), \varphi_{n}(t)\equiv 1 in K (resp. \varphi_{n}(0)\equiv 1 in K_{0} ) and \varphi_{n}(t)\rightarrow0 in W as n\rightarrow \infty (resp. \varphi_{n}(0)\rightarrow0 in H^{1}(\Omega) as n\rightarrow \infty ). In particular \parallel \Delta\varphi_{n}(t)\parallel_{L^{1}(Q)}\rightarrow 0 as n\rightarrow \infty .
Let us consider the nonnegative function \varphi_{n}(t)\in C^{\infty}_{c}(\mathbb{R}^{N}\times \mathbb{R}_{+}) such that \varphi_{n}(T) = 0 in \Omega and \frac{\partial\varphi_{n}(t)}{\partial\eta} = 0 in S as a test function in the problem (P) , then there holds
\begin{equation*} \int_{\Omega}\varphi_{n}(0)du_{0}+\int_{Q}u\varphi_{nt}(t)dxdt = -\int_{Q}\psi(u_{r})\Delta\varphi_{n}(t)dxdt- \int_{Q}h(t)f(x, t)\varphi_{n}(t)dxdt- \end{equation*} |
\begin{equation*} -\int_{S}g(u_{r})\varphi_{n}(t)d\mathcal{H}(x)dt. \end{equation*} | (5.9) |
By (4.3)(Probably n is large enough), the following statement holds
\begin{equation*} \int_{Q}f(x, t)h(t)\varphi_{n}(t)dxdt = \int_{Q}\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma) f(y, \sigma)(h(\sigma)\varphi_{n}(\sigma))_{\sigma}dyd\sigma dxdt \end{equation*} |
\begin{equation*} -\int_{Q}\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)[2\nabla f\nabla(h(\sigma)\varphi_{n}(\sigma))+f\Delta(h(\sigma)\varphi_{n}(\sigma))]dyd\sigma dxdt+ \end{equation*} |
\begin{equation*} +\int_{Q}\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)h(\sigma)\varphi_{n}(\sigma)d\mu(y, \sigma) dxdt+ \end{equation*} |
\begin{equation*} +\int_{Q}\int_{0}^{t}\int_{\partial\Omega}G_{N}(x-y, t-\sigma)h(\sigma)\varphi_{n}(\sigma) g(f(y, \sigma))d\mathcal{H}(y)d\sigma dxdt. \end{equation*} | (5.10) |
Combining the Eq (5.9) with (5.10), we obtain
\begin{equation*} \int_{Q}\int_{\Omega}\int_{0}^{t}G_{N}(x-y, t-\sigma)h(\sigma)\varphi_{n}(\sigma)d\mu(y, \sigma)dxdt+ \int_{\Omega}\varphi_{n}(0)du_{0} = \end{equation*} |
\begin{equation*} = \int_{Q}\int_{0}^{t}\int_{\Omega}h(\sigma)G_{N}(x-y, t-\sigma)[2\nabla f\nabla\varphi_{n}(\sigma)+f\Delta\varphi_{n}(\sigma)]dyd\sigma dxdt- \end{equation*} |
\begin{equation*} -\int_{Q}\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma) f(y, \sigma)(h(\sigma)\varphi_{n}(\sigma))_{\sigma}dyd\sigma dxdt- \end{equation*} |
\begin{equation*} -\int_{Q}\int_{0}^{t}\int_{\partial\Omega}G_{N}(x-y, t-\sigma)h(\sigma)\varphi_{n}(\sigma) g(f(y, \sigma))d\mathcal{H}(y)d\sigma dxdt- \end{equation*} |
\begin{equation*} -\int_{S}\varphi_{n}(t)g(u_{r})d\mathcal{H}(x)dt-\int_{Q}\psi(u_{r})\Delta\varphi_{n}(t)dxdt -\int_{Q}u\varphi_{nt}(t)dxdt. \end{equation*} | (5.11) |
By (2.14), f\in L^{2}((0, T), H^{1}(\Omega)) (see Remark 4.1-(ii)) and letting \sigma\rightarrow t , \int_{0}^{t}h(\sigma)d\sigma = 1 and dropping down the nonnegative terms on the left hand-side of the previous equation. Therefore (5.11) yields
\begin{equation*} \int_{Q}\varphi_{n}(t)d\mu(x, t)+\int_{\Omega}\varphi_{n}(0)du_{0}(x)\leq \parallel u\parallel_{L^{2}((0, T), H^{1}(\Omega))}\parallel \varphi_{n}(t)\parallel_{W}+ \end{equation*} |
\begin{equation*} C(\gamma)\int_{Q}\mid\Delta\varphi_{n}(t)\mid dxdt+\parallel f\parallel_{L^{2}((0, T), H^{1}(\Omega))}\parallel \varphi_{n}(t)\parallel_{W}. \end{equation*} | (5.12) |
Since the following assertions are valid, then
\begin{equation*} \mu^{+}(K)\leq \int_{Q}\varphi_{n}(t)d\mu(x, t)+\int_{Q}\varphi_{n}(t)d\mu^{-}(x, t) \end{equation*} | (5.13) |
where \mu^{+}(K) = \mu(K)+\mu^{-}(K) and
\begin{equation*} u_{0}^{+}(K_{0})\leq \int_{\Omega}\varphi_{n}(0)du_{0}(x)+\int_{\Omega}\varphi_{n}(0)du_{0}^{-}(x) \end{equation*} | (5.14) |
with u_{0}^{+}(K_{0}) = u_{0}(K_{0})+u_{0}^{-}(K_{0}) . In view of (5.13) and (5.14), the inequality (5.12) reads as
\begin{equation*} \mu^{+}(K)+u_{0}^{+}(K_{0})\leq \parallel u\parallel_{L^{2}((0, T), H^{1}(\Omega))}\parallel \varphi_{n}(t)\parallel_{W}+C(\gamma)\int_{Q}\mid\Delta\varphi_{n}(t)\mid dxdt+ \end{equation*} |
\begin{equation*} +\parallel f\parallel_{L^{2}((0, T), H^{1}(\Omega))}\parallel \varphi_{n}(t)\parallel_{W} +\int_{Q}\varphi_{n}(t)d\mu^{-}(x, t)+\int_{\Omega}\varphi_{n}(0)du_{0}^{-}(x). \end{equation*} | (5.15) |
Since \mu^{-}(K) = 0 (resp. u_{0}^{-}(K_{0}) = 0 ), then for any \epsilon > 0 one has
\begin{equation*} \int_{Q}\varphi_{n}(t)d\mu^{-}(x, t) < \frac{\epsilon}{2}\qquad \left(\text{resp.}\quad\int_{\Omega}\varphi_{n}(0)du_{0}^{-}(x) < \frac{\epsilon}{2}\right). \end{equation*} | (5.16) |
Then, the limit in (5.16) as n\rightarrow \infty , the following holds \mu^{+}(K)+u_{0}^{+}(K_{0})\leq\epsilon . Therefore, \mu^{+}(K) = 0 for any compact set K\subset Q (resp. u_{0}^{+}(K_{0}) = 0 for any compact set K_{0}\subset \Omega ).
To prove the existence and decay estimates of the solutions, we consider the following problem
\begin{equation*} \begin{cases} u_{nt} = \Delta\psi_{n}( u_{n})+h(t)f_{n}(x, t) \ \ &\text{in} \ \ Q, \\ \frac{\partial\psi_{n}(u_{n})}{\partial\eta} = g(u_{n}) \ \ &\text{on} \ \ S, \\ u_{n}(x, 0) = u_{0n} \ \ &\text{in} \ \ \Omega, \end{cases} \end{equation*} | (P_{n}) |
where the sequence \{u_{0n}\}\subseteq C^{\infty}_{c}(\Omega) satisfies the assumption (4.1) and the sequence \{f_{n}\}\subseteq C^{\infty}_{c}(\overline{Q}) fulfills the hypothesis (4.19). We set
\begin{equation*} \psi_{n}(s) = \psi(s)+\frac{1}{n} \end{equation*} | (5.17) |
By [8,18,21,22], the approximating problem (P_{n}) has a solution u_{n} in C((0, T), L^{1}(\Omega))\cap L^{\infty}(Q) . Then, the definition of the weak solution \{u_{n}\}\subseteq C^{\infty}(\overline{Q}) of (P_{n}) satisfies the following expression
\begin{equation*} \int_{0}^{T}\langle u_{n}(\cdot, t), \xi_{t}(\cdot, t)\rangle_{\Omega}dt+\int_{0}^{T}h(t)\langle f_{n}(\cdot, t), \xi(\cdot, t)\rangle_{\Omega}dt+ \langle u_{0n}, \xi(\cdot, 0)\rangle_{\Omega}+ \end{equation*} |
\begin{equation*} +\int_{0}^{T}\langle g(u_{n}), \xi\rangle_{\partial\Omega}dt = \int_{0}^{T}\langle\nabla\psi_{n}(u_{n}), \nabla\xi\rangle_{\Omega} dxdt \end{equation*} | (5.18) |
for every \xi in C^{1}(\overline{Q}) such that \xi(\cdot, T) = 0 in \Omega and \frac{\partial\xi}{\partial\eta} = 0 on S .
Now we establish some technical estimates which will be used in the proof of the existing solution.
Lemma 5.2 Assume that (I) , (J) , (A) , \mu\in \mathcal{M}^{+}(Q) and u_{0}\in \mathcal{M}^{+}(\Omega) are satisfied. Let u_{n} be the solution of the approximation problem (P_{n}) , then
\begin{equation*} \parallel u_{n}(\cdot, t)\parallel_{L^{1}(\Omega)} \leq C\left( \parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}\right). \end{equation*} | (5.19) |
\begin{equation*} \parallel\nabla \psi_{n}(u_{n})\parallel_{L^{2}(Q)}+\parallel \psi_{n}(u_{n})\parallel_{L^{2}(Q)}\leq C\left( \parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}\right). \end{equation*} | (5.20) |
for almost every t\in(0, T) and C is a positive constant.
The sequence \left\{[\psi_{n}(u_{n})]_{t}\right\} is bounded in L^{2}((0, T), (H^{1}(\Omega))^{*})+L^{1}(\overline{Q}) .
Proof of Lemma 5.2. To prove the estimate (5.19), we consider the approximation problem (P_{n}) such that
\begin{equation*} \begin{cases} u_{ns} = \Delta\psi_{n}(u_{n})+h(s)f_{n}(x, s) \quad &\text{in} \quad \Omega\times(\tau, \tau+t), \\ \frac{\partial\psi_{n}(u_{n})}{\partial \eta} = g(u_{n}) \quad &\text{on} \quad \partial\Omega\times(\tau, \tau+t), \\ u_{n}(x, \tau) = u_{0n}(x) \quad &\text{in} \quad \Omega\times\{\tau\}, \end{cases} \end{equation*} | (5.21) |
where \tau+t\leq T and \tau , t\in(0, T) .
Let us consider \xi\in C^{1, 2}(\overline{\Omega}\times[\tau, \tau+t]) such that \frac{\partial\xi}{\partial\eta} = 0 on \partial\Omega\times(\tau, \tau+t) and \xi(\cdot, \tau+t) = 0 in \Omega as a test function in the above approximation problem (5.21), then we have
\begin{equation*} \int_{\Omega\times(\tau, \tau+t)}u_{n}\xi_{s}dxds+ \int_{\Omega\times(\tau, \tau+t)}\psi_{n}(u_{n})\Delta \xi dxds+ \int_{\partial\Omega\times(\tau, \tau+t)}g(u_{n})\xi d\mathcal{H}(x)ds+ \end{equation*} |
\begin{equation*} +\int_{\Omega\times(\tau, \tau+t)}h(s)f_{n}(x, s)\xi(x, s) dxds+\int_{\Omega}\mu_{n}(x) \xi(x, \tau)dx = 0. \end{equation*} | (5.22) |
By the mean value theorem and the assumption (I), the Eq (5.22) yields
\begin{equation*} \int_{\Omega\times(\tau, \tau+t)}u_{n}(\xi_{s}+\theta_{n}\Delta \xi)dxds+ \int_{\partial\Omega\times(\tau, \tau+t)}g(u_{n}) \xi d\mathcal{H}(x)ds+ \end{equation*} |
\begin{equation*} +\int_{\Omega\times(\tau, \tau+t)}h(s)f_{n}(x, s) dxds+\int_{\Omega}\mu_{n}(x) \xi(x, \tau)dx = 0, \end{equation*} | (5.23) |
where \theta_{n}(x, t) = \int_{0}^{1}\psi'_{u_{n}}(\alpha u_{n})d\alpha .
On the other hand, we consider the following backward parabolic equations
\begin{equation*} \begin{cases} -\phi_{s}-\theta_{\epsilon}\Delta \phi = \frac{1}{\tau}\quad&\text{in}\, \, \, Q_{\tau} = \Omega\times(\tau, \tau+t), \\ \frac{ \partial \phi}{\partial\eta} = 0\quad&\text{on}\, \, \, S_{\tau} = \partial\Omega\times(\tau, \tau+t), \\ \phi(\cdot, \tau+t) = 0 \quad&\text{in}\, \, \, \Omega\times\{\tau+t\}, \end{cases} \end{equation*} | (5.24) |
has an unique solution \phi in C^{1, 2}(\overline{Q}_{\tau})\cap C(Q_{\tau}) and 0 < \phi\leq C for any \tau, t\in(0, T) (see [18, Lemma 4.2]). Then for \xi = \phi , there holds
\begin{equation*} \frac{1}{\tau}\int_{\tau}^{\tau+t}\int_{\Omega}u_{n}(x, s)dxds = \int_{\Omega}\mu_{n}(x)\phi(x, \tau)dx +\int_{\tau}^{\tau+t}\int_{\partial\Omega}g(u_{n})\phi d\mathcal{H}(x)ds+ \end{equation*} |
\begin{equation*} +\int_{\tau}^{\tau+t}\int_{\Omega}h(s)f_{n}(x, s)\phi dxds \end{equation*} | (5.25) |
By the assumptions (A) , (J) , (4.19) and (4.1), there exists a positive constant C such that the expression below is satisfied
\begin{equation*} \frac{1}{\tau}\int_{\tau}^{\tau+t}\int_{\Omega}u_{n}(x, s)dxds\leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+ \parallel\mu\parallel_{\mathcal{M}^{+}(Q)}). \end{equation*} | (5.26) |
By letting \tau\rightarrow 0^{+} , we obtain the estimate (5.19). Where C = C(h(T), \parallel g(u_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}, \mid S\mid) > 0 . To prove the estimate (5.20), we consider T_{\gamma+1}(\psi_{n}(u_{n})) as a test function in the approximation problem (P_{n}) , then we have
\begin{equation*} \int_{\left\{(x, t)\in Q_{T}/\psi_{n}(u_{n})\leq\gamma+1\right\}}\mid \nabla \psi_{n}(u_{n})\mid^{2}dxdt = \int_{\Omega}\left(\int_{0}^{u_{0n}(x)}T_{\gamma+1}(\psi_{n}(s))ds\right)dx- \end{equation*} |
\begin{equation*} -\int_{\Omega}\left(\int_{0}^{u_{n}(x, T)}T_{\gamma+1}(\psi_{n}(s))ds\right)dx+ \int_{0}^{T}\int_{\partial\Omega}g(u_{n})T_{\gamma+1}(\psi_{n}(u_{n}))d\mathcal{H}(x)dt+ \end{equation*} |
\begin{equation*} +\int_{0}^{T}\int_{\Omega}T_{\gamma+1}(\psi_{n}(u_{n}))h(t)f_{n}(x, t)dxdt \end{equation*} | (5.27) |
where T_{\lambda}(s) = \min\{\lambda, s\} . It follows that there exists a positive constant C such that
\begin{equation*} \int_{\left\{(x, t)\in Q_{T}/\psi_{n}(u_{n})\leq\gamma+1\right\}}\mid \nabla \psi_{n}(u_{n})\mid^{2}dxdt \end{equation*} |
\begin{equation*} \leq (\gamma+1) \int_{\Omega}\mu_{n}(x)dx +C(\gamma+1)\parallel g(u_{n})\parallel_{L^{\infty}( \mathbb{R}_{+})}+h(T)\int_{Q}f_{n}(x, t)dxdt. \end{equation*} |
For the suitable positive constant C = C(h(T), \gamma, \parallel g(u_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}, \parallel\mu\parallel_{\mathcal{M}^{+}(\Omega)}, \parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}, \mid S\mid) > 0 , the following estimate holds
\begin{equation*} \int_{\left\{(x, t)\in Q_{T}/\psi_{n}(u_{n})\leq\gamma+1\right\}}\mid \nabla \psi_{n}(u_{n})\mid^{2}dxdt \leq C. \end{equation*} | (5.28) |
On the other hand, we assume that \mathcal{G}_{\lambda}(s) = \max\{\lambda, s\} and we choose \mathcal{G}_{\gamma+1}(\psi_{n}(u_{n})) as a test function in the approximation problem (P_{n}) , then we have
\begin{equation*} \int_{\left\{(x, t)\in Q_{T}/\psi_{n}(u_{n}) > \gamma+1\right\}}\mid \nabla \psi_{n}(u_{n})\mid^{2}dxdt = \int_{\Omega}\left(\int_{0}^{u_{0n}(x)}\mathcal{G}_{\gamma+1}(\psi_{n}(s)))ds\right)dx- \end{equation*} |
\begin{equation*} -\int_{\Omega}\left(\int_{0}^{u_{n}(x, T)}\mathcal{G}_{\gamma+1}(\psi_{n}(s))ds\right)dx+ \int_{0}^{T}\int_{\partial\Omega}g(u_{n})\mathcal{G}_{\gamma+1}(\psi_{n}(u_{n})) d\mathcal{H}(x)dt+ \end{equation*} |
\begin{equation*} +\int_{0}^{T}\int_{\Omega}h(t)f_{n}(x, t)\mathcal{G}_{\gamma+1}(\psi_{n}(u_{n})) d\mathcal{H}(x)dt. \end{equation*} | (5.29) |
It implies that
\begin{equation*} \int_{\left\{(x, t)\in Q/\psi_{n}(u_{n}) > \gamma+1\right\} }\mid \nabla \psi_{n}(u_{n})\mid^{2}dxdt \end{equation*} |
\begin{equation*} \leq(\gamma+1)\int_{\Omega}\mu_{n}(x)dx +(\gamma+1)M\parallel g(u_{n})\parallel_{L^{\infty}( \mathbb{R}_{+})}+h(T)\int_{Q}f_{n}(x, t)dxdt \end{equation*} |
It follows that
\begin{equation*} \int_{\left\{(x, t)\in Q/\psi_{n}(u_{n}) > \gamma+1\right\} }\mid \nabla \psi_{n}(u_{n})\mid^{2}dxdt\leq C. \end{equation*} | (5.30) |
Combining the inequality (5.28) with (5.30), we deduce that
\begin{equation*} \int_{Q}\mid \nabla \psi_{n}(u_{n})\mid^{2}dxdt\leq C \end{equation*} | (5.31) |
By the assumption (I), then \psi_{n}(u_{n})\in L^{2}(Q) , whence the estimate (5.20) holds.
To end the proof of this Lemma, we consider that for every \xi\in C^{1}_{c}(Q) such that if we choose \phi = \psi'_{n}(u_{n})\xi arbitrary as a test function in problem (P_{n}) , then the following stands true
\begin{equation*} \int_{Q}\xi_{t}[\psi_{n}(u_{n})]dxdt = -\int_{Q}\xi \psi'_{n}(u_{n})\text{div}\left(\nabla\psi_{n}(u_{n})\right)dxdt -\int_{Q}h(t)f_{n}(x, t)\psi'_{n}(u_{n})\xi dxdt \end{equation*} | (5.32) |
It follows that
\begin{equation*} \int_{Q}\xi_{t}[\psi_{n}(u_{n})]dxdt = \int_{Q}\psi_{n}'(u_{n})\nabla\psi_{n}(u_{n})\nabla\xi dxdt-\int_{Q}h(t)f_{n}(x, t)\phi dxdt- \end{equation*} |
\begin{equation*} -\int_{S}g(u_{n})\psi_{n}'(u_{n})\xi d\mathcal{H}(x)dt+\int_{Q}\frac{\psi_{n}''(u_{n})}{\psi_{n}'(u_{n})}\mid \nabla\psi_{n}(u_{n})\mid^{^{2}}\xi dxdt. \end{equation*} | (5.33) |
Now we estimate each term in the right hand side of (5.33), we obtain
\begin{equation*} \left|\int_{Q}\psi_{n}'(u_{n})\nabla\psi_{n}(u_{n})\nabla\xi dxdt\right|\leq \parallel \psi_{n}'\parallel_{L^{\infty}( \mathbb{R}_{+})}\int_{Q}\mid\nabla\xi\mid\mid\nabla\psi_{n}(u_{n})\mid dxdt. \end{equation*} | (5.34) |
From Hölder's inequality and (5.31), the inequality (5.34) reads as
\begin{equation*} \left|\int_{Q}\psi_{n}'(u_{n})\nabla\psi_{n}(u_{n})\nabla\xi dxdt\right|\leq C\parallel \nabla\xi\parallel_{L^{2}(Q)}. \end{equation*} | (5.35) |
By the assumption (J) and (4.19), we deduce the estimate
\begin{equation*} \left|\int_{Q}h(t)f_{n}(x, t)\xi dxdt\right|\leq C\parallel\xi\parallel_{L^{\infty}(Q)} \end{equation*} | (5.36) |
where C = C(h(T), \parallel\mu\parallel_{\mathcal{M}^{+}(\Omega)}, \parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}) > 0 is a constant.
By the assumptions (A) and (I) , there exists a positive constant C = C(\parallel g(u_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}, \parallel \psi'_{n}(u_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}) > 0 such that
\begin{equation*} \int_{S}g(u_{n})\psi_{n}'(u_{n})\xi d\mathcal{H}(x)dt\leq C\parallel\xi\parallel_{L^{\infty}(S)}. \end{equation*} | (5.37) |
Furthermore, one has
\begin{equation*} \left|\int_{Q}\frac{\psi_{n}''(u_{n})}{\psi_{n}'(u_{n})}\mid \nabla\psi_{n}(u_{n})\mid^{^{2}}\xi dxdt\right| \leq \kappa\parallel\xi\parallel_{L^{\infty}(Q)} \int_{Q}\mid\nabla \psi_{n}(u_{n})\mid^{2} dxdt. \end{equation*} |
In view of (5.28), the expression below holds true
\begin{equation*} \left|\int_{Q}\frac{\psi_{n}''(u_{n})}{\psi_{n}'(u_{n})}\mid \nabla\psi_{n}(u_{n})\mid^{^{2}}\xi dxdt\right|\leq C\parallel\xi\parallel_{L^{\infty}(Q)} \end{equation*} | (5.38) |
where C = C(\kappa, \parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}, \parallel \mu\parallel_{\mathcal{M}^{+}(Q)}) > 0 . By (5.35)-(5.38) and (5.33), we infer that the sequence \left\{[\psi_{n}(u_{n})]_{t}\right\} is bounded in L^{2}((0, T), (H^{1}(\Omega))^{*})+L^{1}(\overline{Q}) .
Now we study the limit points of the sequences \{u_{n}\} and \psi_{n}(u_{n}) as n\rightarrow \infty .
Proposition 5.1 Suppose that the assumptions (I) , (A) and (J) are satisfied. Let u_{n} be the solution of the approximation problem (P_{n}) . Then there exists a subsequence \{u_{n_{k}}\}\subseteq\{ u_{n}\} and v\in L^{2}((0, T), H^{1}(\Omega))\cap L^{\infty}(Q) such that
\begin{equation*} \psi_{n_{k}}(u_{n_{k}})\overset{*}\rightharpoonup v\quad\text{in}\quad L^{\infty}(Q). \end{equation*} | (5.39) |
\begin{equation*} \psi_{n_{k}}(u_{n_{k}})\rightharpoonup v\quad\text{in}\quad L^{2}((0, T), H^{1}(\Omega)). \end{equation*} | (5.40) |
\begin{equation*} [\psi_{n_{k}}(u_{n_{k}})]_{t}\rightharpoonup v_{t}\quad\text{in}\quad L^{2}((0, T), (H^{1}(\Omega))^{*}). \end{equation*} | (5.41) |
\begin{equation*} \psi_{n_{k}}(u_{n_{k}})\rightarrow v\quad\text{a.e in}\quad Q, \end{equation*} | (5.42) |
where v_{t}\in L^{1}(Q) and v\leq \gamma .
Proof of Proposition 5.1. The convergences (5.39) and (5.40) are the consequence of assumption (I)-(i) and estimate (5.20) respectively. By Lemma 5.1, the sequence \left\{[\psi_{n}(u_{n})]_{t}\right\} is bounded in L^{2}((0, T), (H^{1}(\Omega))^{*})+L^{1}(\overline{Q}) . By [45], there exists a subsequence \{u_{n_{k}}\}\subseteq\{ u_{n}\} and v^{*}\in L^{2}((0, T), H^{1}(\Omega))\cap L^{\infty}(\overline{Q}) such that
\begin{equation*} \psi_{n_{k}}(u_{n_{k}})\rightarrow v^{*}\quad\text{a.e in}\quad Q. \end{equation*} |
Furthermore, by [9, Proposition 5.1] and (5.41) holds true and we have
\begin{equation*} \psi_{n_{k}}(u_{n_{k}})\rightarrow v\quad\text{a.e in}\quad Q \end{equation*} |
with v = v^{*} which leads to (5.42) be satisfied. In view of the assumptions (I)- (i) and (5.17), we get
\begin{equation*} \parallel \psi_{n_{k}}(u_{n_{k}})-\psi(u_{n_{k}})\parallel_{L^{\infty}(Q)} = \frac{1}{n_{k}}. \end{equation*} |
Therefore the following convergence \psi(u_{n_{k}})\overset{*}\rightharpoonup v\quad\text{in}\quad L^{\infty}(Q) holds true.
Remark 5.1 For any subsequence \{u_{n_{k}}\}\subseteq \{u_{n}\} and v the function given in Proposition 5.1, the following assertions
\psi^{-1}(v)\in L^{\infty}((0, T), L^{1}(\Omega)) , u_{n_{k}}\rightarrow \psi^{-1}(v)\quad\text{a.e in }\; \; Q and u_{n_{k}}\rightarrow g^{-1}(v)\quad\text{a.e in }\; \; S hold.
Proposition 5.2 Assume that the hypotheses (I) , (J) , (A) , \mu\in \mathcal{M}^{+}(Q) and u_{0}\in \mathcal{M}^{+}(\Omega) are satisfied. Let \{u_{n_{k}}\} be the subsequence and v the function mentioned in Proposition 5.1. Then there exist a subsequence \{u_{n_{k}}(\cdot, t)\}\subseteq \{u_{n}(\cdot, t)\} and u_{a}\;, \; u(\cdot, t)\;, \; u_{b}(\cdot, t)\in \mathcal{M}^{+}(\Omega) such that
\begin{equation*} u_{n_{k}}(\cdot, t)\overset{*}\rightharpoonup u(\cdot, t): = u_{a}(\cdot, t)+u_{b}(\cdot, t)\quad\text{in}\quad \mathcal{M}^{+}(\Omega), \end{equation*} | (5.43) |
\begin{equation*} \psi_{n_{k}}(u_{n_{k}})(\cdot, t)\overset{*}\rightharpoonup \psi(u_{b})(\cdot, t)\quad\text{in}\quad \mathcal{M}^{+}(\Omega), \end{equation*} | (5.44) |
\begin{equation*} g(u_{n_{k}})(\cdot, t)\overset{*}\rightharpoonup g(u_{b})(\cdot, t)\quad\text{in}\quad L^{\infty}(\partial\Omega). \end{equation*} | (5.45) |
Moreover, there hold
\begin{equation*} u_{b}(\cdot, t) = u_{r}(\cdot, t)\;\;\text{a.e in}\;\;\Omega\;\;\text{and}\;\;u_{a}(\cdot, t) = u_{s}(\cdot, t)\;\;\text{in}\quad\mathcal{M}^{+}(\Omega) \end{equation*} | (5.46) |
for almost every t\in(0, T) . Furthermore u\in L^{\infty}((0, T), \mathcal{M}^{+}(\Omega)) and for almost every t\in (0, T) , there holds
\begin{equation*} \parallel u(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq C(\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}+\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}). \end{equation*} | (5.47) |
Proof. By the assumption (I)- (i) , \psi_{n_{k}}(u_{n_{k}})\in L^{\infty}(Q) and using Hölder's inequality, we have
\begin{equation*} \int_{Q}\mid \nabla\psi_{n_{k}}(u_{n_{k}})\mid dxdt\leq\left[\int_{Q}\frac{\mid \nabla\psi_{n_{k}}(u_{n_{k}})\mid^{2}}{(1+\psi_{n_{k}}(u_{n_{k}}))^{2}}dxdt\right]^{\frac{1}{2}} \left[\int_{Q}(1+\psi_{n_{k}}(u_{n_{k}}))^{2}dxdt\right]^{\frac{1}{2}} \end{equation*} |
\begin{equation*} \leq C \left[\int_{Q}\mid \nabla\psi_{n_{k}}(u_{n_{k}})\mid^{2}dxdt\right]^{\frac{1}{2}}. \end{equation*} |
From the estimate (5.20), there exists a positive constant C = C(\parallel\psi_{n_{k}}(u_{n_{k}}) \parallel_{L^{\infty}(\mathbb{R}_{+})}, \parallel\mu \parallel_{\mathcal{M}^{+}(Q)}, \parallel u_{0} \parallel_{\mathcal{M}^{+}(\Omega)}) > 0 such that
\begin{equation*} \int_{Q}\mid \nabla\psi_{n_{k}}(u_{n_{k}})\mid dxdt\leq C. \end{equation*} | (5.48) |
According to Lemma 5.1, the assumption (I) and (5.48), we infer that
\begin{equation*} \parallel\psi_{n_{k}}(u_{n_{k}})\parallel_{BV(Q)} = \parallel\psi_{n_{k}}(u_{n_{k}})\parallel_{L^{1}(Q)}+ \parallel\nabla\psi_{n_{k}}(u_{n_{k}})\parallel_{L^{1}(Q)} +\parallel[\psi_{n_{k}}(u_{n_{k}})]_{t}\parallel_{L^{1}(Q)}\leq C. \end{equation*} | (5.49) |
By Fatou's Lebesgue Lemma, we obtain
\begin{equation*} \int_{0}^{T}\liminf\limits_{k\rightarrow \infty}\int_{\Omega}\left\{\mid \psi_{n_{k}}(u_{n_{k}})\mid +\mid \nabla\psi_{n_{k}}(u_{n_{k}})\mid+\mid [\psi_{n_{k}}(u_{n_{k}})]_{t}\mid\right\}\leq C. \end{equation*} | (5.50) |
Then there exists zero Lebesgue measure set \mathcal{N}_{1}\subset(0, T) such that
\begin{equation*} \liminf\limits_{k\rightarrow \infty}\int_{\Omega}\left\{\mid \psi_{n_{k}}(u_{n_{k}})\mid +\mid \nabla\psi_{n_{k}}(u_{n_{k}})\mid+\mid [\psi_{n_{k}}(u_{n_{k}})]_{t}\mid\right\}(x, t)\leq C \end{equation*} | (5.51) |
for every t\in(0, T)\backslash \mathcal{N}_{1} . In view of (5.51), the sequence \{\psi_{n_{k}}(u_{k})(\cdot, t)\}\subseteq BV(\Omega) for every t\in(0, T)\backslash \mathcal{N}_{1} . By [44, Chapter IV, Section 1.1, Proposition 5], there exists a subsequence \{\psi_{n_{k}}(u_{n_{k}})(\cdot, t)\} and v(\cdot, t)\in\mathcal{M}^{+}(\Omega) a.e in (0, T) such that the convergence
\begin{equation*} \psi_{n_{k}}(u_{n_{k}})(\cdot, t)\overset{*}\rightharpoonup v(\cdot, t)\quad\text{in}\quad \mathcal{M}^{+}(\Omega) \end{equation*} | (5.52) |
holds true. Furthermore, from the assertions (5.19), (5.52) and the Prohorov Theorem (see [44, Chapter II, Section 2.6, Theorem 1] or [25, Proposition A.2] or [17, Proposition 1]), there exists a sequence \{\widetilde{\tau}^{n_{k}}\} of the Young measures associated with the sequence \{u_{n_{k}}\}\subseteq \{u_{n}\} converges narrowly over \overline{Q}\times \mathbb{R} to a Young measure \widetilde{\tau} which the disintegration \overline{\mu}_{(\cdot, t)} is the Dirac mass concentrated at the point \psi^{-1}(v(\cdot, t)) for a.e in \Omega (see [17]). By [25, Proposition A.4], there exist sequences of measure sets A_{k}\subseteq \Omega , A_{k}\subseteq A_{k+1} and \mid A_{k}\mid\rightarrow 0 , such that
\begin{equation*} u_{k_{j}}(\cdot, t)\chi_{\Omega\backslash A_{k}}\rightharpoonup u_{b}(\cdot, t): = \int_{[0, +\infty)}\lambda d\overline{\mu}_{(\cdot, t)}(\lambda)\quad\text{in}\quad L^{1}(\Omega), \end{equation*} | (5.53) |
where u_{b}\in L^{\infty}((0, T), L^{1}(\Omega)) , u_{b}\geq 0 is a barycenter of the limiting Young measure \overline{\mu}_{(\cdot, t)} associated with the subsequence \{u_{n_{k}}(\cdot, t)\} and supp \overline{\mu}_{(\cdot, t)}\subseteq[0, +\infty) for almost every t\in(0, T) .
By (5.19) and the compactness result, the sequence \{u_{n_{k}}\chi_{\Omega\backslash A_{j}}\} is uniformly bounded in L^{1}(\Omega) . Therefore, there exists a Radon measure u_{a}(\cdot, t)\in \mathcal{M}^{+}(\Omega) such that u_{n_{k}}(\cdot, t)\overset{*}\rightharpoonup u(\cdot, t)\; \; \text{ in }\; \, \mathcal{M}^{+}(\Omega) . Finally, the sequence u_{n_{k}} is of u_{n_{k}}(\cdot, t) = u_{n_{k}}(\cdot, t)\chi_{A_{k}}+u_{n_{k}}(\cdot, t)\chi_{\Omega\backslash A_{k}}\overset{*}\rightharpoonup u_{a}(\cdot, t)+ u_{b}(\cdot, t)\; \; \text{ in }\:\: \mathcal{M}^{+}(\Omega) . Hence u(\cdot, t): = u_{a}(\cdot, t)+u_{b}(\cdot, t) in \mathcal{M}^{+}(\Omega) and the statement (5.43) is completed. By the assumption (I)- (iii) , there holds
\begin{equation*} \lim\limits_{_{s\rightarrow +\infty}}\frac{\psi_{n_{k}}(s)}{s} = 0. \end{equation*} | (5.54) |
By the assertion (5.54) and [45, Proposition 5.2] or [25], we obtain
\begin{equation*} \psi_{n_{k}}(u_{k})(\cdot, t)\overset{*}\rightharpoonup\psi^{*}(\cdot, t)\quad\text{in}\quad \mathcal{M}^{+}(\Omega) \end{equation*} | (5.55) |
where \psi^{*}(\cdot, t)\in L^{1}(\Omega) and
\begin{equation*} \psi^{*}(\cdot, t) = \int_{[0, +\infty)}\psi(\lambda) d\overline{\mu}_{(\cdot, t)}(\lambda). \end{equation*} | (5.56) |
Furthermore, we also obtain the next result via (5.55)
\begin{equation*} \psi^{*}(\cdot, t) = \int_{[0, +\infty)}\psi(\lambda) d\overline{\mu}_{(\cdot, t)}(\lambda) = \psi\left(\int_{[0, +\infty)}\lambda d\overline{\mu}_{(\cdot, t)}(\lambda)\right) = \psi(u_{b})(\cdot, t). \end{equation*} |
By combining the assertion (5.53) and the previous equality, we conclude that \psi(u_{b})(\cdot, t) = v(\cdot, t) a.e in (0, T) , when the convergence (5.44) is satisfied.
By virtue of the convergence (5.53), the next convergence result
\begin{equation*} g(u_{n_{k}})\rightarrow g(\psi^{-1}(v)): = g(u_{b})\quad\text{a.e in}\quad S \end{equation*} | (5.57) |
holds true. Since the function g(u_{n_{k}})\in L^{\infty}(\mathbb{R}_{+}) (see assumption (H)-(i)) and from Fatou's Lebesgue Lemma, then there exists a positive constant C such that
\begin{equation*} \int_{0}^{T}\liminf\limits_{k\rightarrow +\infty}\int_{\partial\Omega}g(u_{n_{k}})dxdt\leq C. \end{equation*} | (5.58) |
Therefore, there exists a zero Lebesgue measure set \mathcal{N}_{2}\subseteq(0, T) such that
\begin{equation*} \liminf\limits_{k\rightarrow +\infty}\int_{\partial\Omega}g(u_{n_{k}})(x, t)dx\leq C \end{equation*} | (5.59) |
for every t\in(0, T)\backslash \mathcal{N}_{2} . In view of (5.59) and (5.57), there exists a function z(\cdot, t): = g(u_{b})(\cdot, t)\in L^{\infty}(\partial\Omega) such that the convergence (5.45) is achieved.
To show (5.46), we consider the functions \mathrm{F}\;, \; \mathrm{G}: \mathbb{R}_{+}\rightarrow \mathbb{R}_{+} defined by setting
\begin{equation*} \mathrm{F}_{\epsilon}(s) = \begin{cases} 0 \ \ &\text{if} \ \ \ s\leq \frac{1}{\epsilon}, \\ \frac{(\epsilon s-1)^{2}}{2\epsilon^{2}} \ \ &\text{if} \ \ \frac{1}{\epsilon}\leq s\leq \frac{1}{\epsilon}+1, \\ s-\frac{1}{\epsilon}-\frac{1}{2} \ \ &\text{if} \ \ s\geq \frac{1}{\epsilon}+1, \end{cases} \end{equation*} |
and \mathrm{G}_{\epsilon}(s) = s-\mathrm{F}_{\epsilon}(s) for every \epsilon > 0 . It is worthy observing that \mathrm{F}'_{\epsilon}(s)\geq 0 in \mathbb{R}_{+} and 0\leq\mathrm{F}''_{\epsilon}(s)\leq\chi_{\{s\geq\frac{1}{\epsilon}\}}(s) . According to the above results, there exists a subsequence \{u_{n_{k}}\} in Lemma 5.1 and Proposition 5.1. For any nonnegative function \rho\in C^{2}(\overline{\Omega}) , we choose \mathrm{F}'_{\epsilon}(u_{n_{k}})\rho(x) as a test function in the approximation problem (P_{n}) , then we obtain the following identity
\begin{equation*} \int_{\Omega}\mathrm{F}_{\epsilon}(u_{n_{k}})(\cdot, \tau)\rho(x)dx\leq\int_{\Omega} \mathrm{F}_{\epsilon}(u_{0n_{k}})\rho(x)dx- \int_{0}^{\tau}\int_{\Omega}\mathrm{F}'_{\epsilon}(u_{n_{k}})\nabla\psi_{n_{k}}(u_{n_{k}})\nabla\rho(x)dxdt+ \end{equation*} |
\begin{equation*} +\int_{0}^{\tau}\int_{\partial\Omega}g(u_{n_{k}}) \mathrm{F}'_{\epsilon}(u_{n_{k}})\rho(x)d\mathcal{H}(x)dt+\int_{0}^{\tau}\int_{\Omega}h(t)f_{n_{k}}(x, t) \mathrm{F}'_{\epsilon}(u_{n_{k}})\rho(x)dxdt \end{equation*} | (5.60) |
where \tau\in(0, T) . Since the sequence \{\mathrm{F}'_{\epsilon}(u_{n_{k}})\} is uniformly bounded in L^{\infty}(Q) , then \mathrm{F}'_{\epsilon}(u_{n_{k}})\rightarrow0 as \epsilon\rightarrow 0^{+} and \mathrm{F}_{\epsilon}(u_{n_{k}})\rightarrow0 as \epsilon \rightarrow0^{+} . By Lemma 5.1 and Proposition 5.1, and by applying the Dominated Convergence Theorem, results to
\begin{equation*} \lim\limits_{k\rightarrow +\infty}\int_{0}^{\tau}\int_{\Omega} \mathrm{F}'_{\epsilon}(u_{n_{k}})\nabla\psi_{n_{k}}(u_{n_{k}})\nabla\rho(x)dxdt = \int_{0}^{\tau}\int_{\Omega}\mathrm{F}'_{\epsilon}(\psi^{-1}(v))\nabla v\nabla\rho(x)dxdt. \end{equation*} | (5.61) |
Similary, we get
\begin{equation*} \lim\limits_{k\rightarrow +\infty}\int_{0}^{\tau}\int_{\partial\Omega}g(u_{n_{k}}) \mathrm{F}'_{\epsilon}(u_{n_{k}})\rho(x)d\mathcal{H}(x) dydt = \int_{0}^{\tau}\int_{\partial\Omega}g(\psi^{-1}(v)) \mathrm{F}'_{\epsilon}(\psi^{-1}(v))\rho(x)d\mathcal{H}(x)dt, \end{equation*} | (5.62) |
By the statement (4.19) and Proposition 5.1, we have
\begin{equation*} \lim\limits_{k\rightarrow +\infty}\int_{0}^{\tau}\int_{\Omega}h(t)f_{n_{k}}(x, t) \mathrm{F}'_{\epsilon}(u_{n_{k}})\rho(x)dxdt = \int_{0}^{\tau}\int_{\Omega}h(t)f(x, t) \mathrm{F}'_{\epsilon}(\psi^{-1}(v))\rho(x)d\mathcal{H}(x)dt. \end{equation*} | (5.63) |
Given the properties of the sequence \{\mathrm{F}_{\epsilon}(u_{n_{k}})\} and passing to limit in (5.61), (5.62) and (5.63) when \epsilon\rightarrow 0^{+} , then the following holds
\begin{equation*} \lim\limits_{\epsilon\rightarrow 0^{+}}\lim\limits_{k\rightarrow +\infty}\int_{0}^{\tau}\int_{\Omega} \mathrm{F}'_{\epsilon}(u_{n_{k}})\nabla\psi_{n_{k}}(u_{n_{k}})\nabla\rho(x)dxdt = 0. \end{equation*} | (5.64) |
Similarly we obtain
\begin{equation*} \lim\limits_{\epsilon\rightarrow 0^{+}}\lim\limits_{k\rightarrow +\infty}\int_{0}^{\tau}\int_{\partial\Omega}g(u_{n_{k}}) \mathrm{F}'_{\epsilon}(u_{n_{k}})\rho(x)d\mathcal{H}(x)dt = 0. \end{equation*} | (5.65) |
And
\begin{equation*} \lim\limits_{\epsilon\rightarrow 0^{+}}\lim\limits_{k\rightarrow +\infty}\int_{0}^{\tau}\int_{\Omega}h(t)f_{n_{k}}(x, t) \mathrm{F}'_{\epsilon}(u_{n_{k}})\rho(x)dxdt = 0. \end{equation*} | (5.66) |
On the other hand, we have
\begin{equation*} \mathrm{F}_{\epsilon}(u_{0n_{k}}) = u_{0n_{k}}-\mathrm{G}_{\epsilon}(u_{0n_{k}}) = u_{0rn_{k}}+u_{0sn_{k}} -\mathrm{G}_{\epsilon}(u_{0n_{k}}). \end{equation*} |
Since u_{0rn_{k}}\rightarrow u_{0r} in L^{1}(\Omega) , u_{0sn_{k}}\overset{*}\rightharpoonup u_{0s} in \mathcal{M}^{+}(\Omega) and the sequence \{\mathrm{G}_{\epsilon}(u_{0n_{k}})\} is uniformly bounded in L^{\infty}(\Omega) , then we deduce that
\begin{equation*} u_{0rn_{k}}-\mathrm{G}_{\epsilon}(u_{0n_{k}})\rightarrow u_{0r}-\mathrm{G}_{\epsilon}(u_{0r}): = \mathrm{F}_{\epsilon}(u_{0r})\quad\text{in}\quad L^{1}(\Omega). \end{equation*} | (5.67) |
According to the convergence statement (5.43), we have
\begin{equation*} \mathrm{F}_{\epsilon}(u_{n_{k}})(\cdot, t) = u_{n_{k}}(\cdot, t)-\mathrm{G}_{\epsilon}(u_{n_{k}})(\cdot, t) \overset{*}\rightharpoonup u_{a}(\cdot, t)+\psi^{-1}(v)-\mathrm{G}_{\epsilon}(\psi^{-1}(v))(\cdot, t)\;\;\text{in}\quad \mathcal{M}^{+}(\Omega) \end{equation*} | (5.68) |
where \mathrm{F}_{\epsilon}(\psi^{-1}(v))(\cdot, t): = \psi^{-1}(v)(\cdot, t)-\mathrm{G}_{\epsilon}(\psi^{-1}(v))(\cdot, t) .
Furthermore, from the Eqs (5.43) and (5.66) we obtain the following
\begin{equation*} \lim\limits_{\epsilon\rightarrow0^{+}}\lim\limits_{k\rightarrow +\infty} \int_{\Omega}\mathrm{F}_{\epsilon}(u_{n_{k}})(\cdot, t)\rho(x)dx = \langle u_{a}(\cdot, t), \rho\rangle_{\Omega}+\lim\limits_{\epsilon\rightarrow0^{+}} \int_{\Omega}\mathrm{F}_{\epsilon}(\psi^{-1}(v))(\cdot, t)\rho(x)dx. \end{equation*} | (5.69) |
It follows that
\begin{equation*} \lim\limits_{\epsilon\rightarrow0^{+}}\lim\limits_{k\rightarrow +\infty} \int_{\Omega}\mathrm{F}_{\epsilon}(u_{n_{k}})(\cdot, t)\rho(x)dx = \langle u_{a}(\cdot, t), \rho\rangle_{\Omega}. \end{equation*} | (5.70) |
Likewise, from (5.67) one has
\begin{equation*} \lim\limits_{\epsilon\rightarrow0^{+}}\lim\limits_{k\rightarrow +\infty} \int_{\Omega}\mathrm{F}_{\epsilon}(u_{0n_{k}})(\cdot, t)\rho(x)dx = \langle u_{0s}, \rho\rangle_{\Omega} +\lim\limits_{\epsilon\rightarrow0^{+}}\int_{\Omega}\mathrm{F}_{\epsilon}(u_{0r})\rho(x)dx. \end{equation*} |
It implies that
\begin{equation*} \lim\limits_{\epsilon\rightarrow0^{+}}\lim\limits_{k\rightarrow +\infty} \int_{\Omega}\mathrm{F}_{\epsilon}(u_{0n_{k}})\rho(x)dx = \langle u_{0s}, \rho\rangle_{\Omega}. \end{equation*} | (5.71) |
Combining the statements (5.64)-(5.66), (5.70), (5, 71) with (5.60) yields
\begin{equation*} \langle u_{a}(\cdot, t), \rho\rangle_{\Omega}\leq\langle u_{0s}, \rho\rangle_{\Omega}. \end{equation*} |
Since u_{a}(\cdot, t) is a singular measure with respect to the Lebesgue measure u_{a}(\cdot, t) = [u_{a}(\cdot, t)]_{s} = u_{s}(\cdot, \overline{t}) for a suitable \overline{t}\in (0, T)\setminus H^{*} , where H^{*} is zero Lebesgue measure in (0, T) . Hence the assertion (5.46) is obtained.
From [44, Chapter 5, Section 5.2.1, Theorem 1], the estimate (5.19) yields
\begin{equation*} \parallel u(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq\liminf\limits_{k\rightarrow +\infty}\int_{\Omega} u_{n_{k}}(\cdot, t)dx\leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}). \end{equation*} | (5.72) |
The estimate (5.47) is completed.
Proof of Theorem 2.2. By Proposition 5.1 and Proposition 5.2, we have \psi(u_{r}) = v a.e in Q . Hence the problem (P) has a weak Radon measure-valued solution u in L^{\infty}((0, T), \mathcal{ M}^{+}(\Omega)) .
Remark 5.1 By Theorem 2.2, the result holds
\begin{equation*} [u(\cdot, t)]_{s}\leq u_{0s}\quad\text{in}\quad \mathcal{M}^{+}(\Omega) \end{equation*} | (5.73) |
for almost every t\in(0, T) . By (5.73), there exists zero Lebesgue measure set \mathcal{N}_{3}\subset(0, T) such that
\begin{equation*} [u(\cdot, t)]_{c, 2}(E)\leq [u_{0}]_{c, 2}(E)\quad\text{in}\quad \mathcal{M}^{+}(\Omega) \end{equation*} | (5.74) |
for all Borel sets E\subset\Omega , with C_{2}(E) = 0 and t\in(0, T)\backslash\mathcal{N}_{3} .
Proposition 5.3. Suppose that the assumptions (I) and (A) are fulfilled. Let \{u_{n_{k}}\} be the subsequence and v the function given in Proposition 5.1. Then the following sets
\begin{equation*} \mathcal{S} = \left\{(x, t)\in \overline{Q}\setminus \psi(u_{r})(x, t) = \gamma\right\}\;\text{and}\;\mathcal{N} = \left\{(x, t)\in \overline{Q}\setminus g(u_{r})(x, t) = 0\right\} \end{equation*} |
have zero Lebesgue measure. Moreover \mathcal{S}\subseteq \mathcal{N} and \mathcal{B} = \mathcal{S}\cup\mathcal{N} has zero Lebesgue measure.
Proof of Proposition 5.3. By [9, Proposition 5.2], the set \mathcal{S} has zero Lebesgue measure. Assume that
\begin{equation*} A_{j} = \left\{(x, t)\in \overline{Q}\setminus v(x, t)\leq\frac{1}{j}\right\}. \end{equation*} |
Then, it is worth observing that
\begin{equation*} A_{j+1}\supseteq A_{j}\;, \;\mathcal{N} = \bigcup\limits_{j = 1}^{\infty}A_{j}\;, \;\mid \mathcal{N}\mid = \lim\limits_{j\rightarrow +\infty} \mid A_{j}\mid \end{equation*} | (5.75) |
To prove that \mid\mathcal{N}\mid = 0 , it is enough to show that \mid A_{j}\mid \rightarrow0 as j\rightarrow +\infty .
Since the function g' < 0 in \mathbb{R}_{+} (see the assumption (A) -(i)), then we have
\begin{equation*} g(u_{n_{k}})\leq\frac{2}{j}\Leftrightarrow u_{n_{k}}\geq g^{-1}\left(\frac{2}{j}\right)\;((x, t)\in \overline{Q}). \end{equation*} | (5.76) |
It follows that
\begin{equation*} g^{-1}\left(\frac{2}{j}\right)\int_{\left\{(x, t)\in \overline{Q}\setminus v(x, t)\leq\frac{1}{j}\right\}} \chi_{\left\{g(u_{n_{k}})\leq\frac{2}{j}\right\}}dxdt\leq \int_{Q}u_{n_{k}}(x, t)dxdt. \end{equation*} | (5.77) |
By the estimate (5.19), we have
\begin{equation*} g^{-1}\left(\frac{2}{j}\right)\mid A_{j}\mid\leq CT\parallel\mu\parallel_{\mathcal{M}^{+}(\Omega)}. \end{equation*} | (5.78) |
Since g^{-1}\left(\frac{2}{j}\right)\rightarrow +\infty as j\rightarrow +\infty , then (3.62) yields \mid A_{j}\mid \rightarrow0 as j\rightarrow +\infty .
Assume that (x_{0}, t_{0})\in \mathcal{S} , then \psi(u_{r}(x_{0}, t_{0})) = \gamma for every \gamma\in(0, +\infty) . Since g(u_{r}(x_{0}, t_{0})) = \frac{\partial\psi(u_{r})}{\partial\eta}(x_{0}, t_{0}) = \frac{\partial}{\partial\eta}(\gamma) = 0 . Therefore, (x_{0}, t_{0}))\in \mathcal{N} , that is \mathcal{S}\subseteq\mathcal{N} holds true. The fact that \mathcal{S}\subseteq\mathcal{N} , then \mathcal{B} = \mathcal{N} . Consequently, \mathcal{B} is zero Lebesgue measure set.
Proposition 6.1. Under assumptions (I) , (A) and (J) . Let u be a very weak Radon measure-valued solution to the problem (P) and for every \rho\in C^{2}(\overline{\Omega}) such that \frac{\partial\rho}{\partial\eta} = 0 on \partial\Omega , there holds
\begin{equation*} \text{ess}\lim\limits_{t\rightarrow0^{+}}\langle u(\cdot, t), \rho\rangle_{\Omega} = \langle u_{0}, \rho\rangle_{\Omega} \end{equation*} | (6.1) |
Proof of Proposition 6.1 Let us consider that for every \tau > 0 , the smooth function \eta_{\tau}\in C^{1}_{c}(0, T) , 0\leq \eta_{\tau}\leq 1 such that
\begin{equation*} \eta_{\tau}(t) = \begin{cases} 0\quad&\text{if}\, \, \, 0\leq t\leq t_{1}-\tau, \\ \frac{1}{\tau}\left(t+\tau-t_{1}\right)\quad&\text{if}\, \, \, t_{1}-\tau\leq t\leq t_{1}, \\ 1\quad&\text{if}\, \, \, t_{1}\leq t\leq t_{2}, \\ \frac{1}{\tau}\left(-t+\tau+t_{2}\right)\quad&\text{if}\, \, \, t_{2}\leq t\leq t_{2}+\tau, \\ 0\quad&\text{if}\, \, \, t_{2}+\tau\leq t\leq T. \end{cases} \end{equation*} | (6.2) |
Let us choose \rho_{j}(x)\eta_{\tau}(t) as a test function in (P) , there holds
\begin{equation*} \int_{0}^{T}\int_{\Omega}\left\{-u\rho_{j}(x)\eta'_{\tau}(t)-\psi(u_{r})\eta_{\tau}(t)\Delta\rho_{j}(x) \right\}dxdt = \end{equation*} |
\begin{equation*} \int_{0}^{T}\int_{\partial\Omega}g(u_{r})\rho_{j}(x)\eta_{\tau}(t)d\mathcal{H}(x)dt +\int_{0}^{T}\int_{\Omega}h(t)f\eta_{\tau}(t)\rho_{j}(x)dxdt. \end{equation*} | (6.3) |
It is worth observing that the first term on the left hand side of the equality (6.3) gives
\int_{0}^{T}\int_{\Omega}-u\rho_{j}(x)\eta'_{\tau}(t)dxdt = -\frac{1}{\tau}\int_{t_{1} -\tau}^{t_{1}}\int_{\Omega}u(x, t)\rho_{j}(x)dxdt+ |
\begin{equation*} +\frac{1}{\tau}\int_{t_{2}}^{t_{2}+\tau}\int_{\Omega}u(x, t)\rho_{j}(x)dxdt. \end{equation*} | (6.4) |
Let us consider a zero Lebesgue measure set D_{j} in (0, T) such that for any t_{1}, \; t_{2}\in (0, T)\setminus D_{j} , one has
\begin{equation*} \lim\limits_{\tau\rightarrow0}\int_{0}^{T}\int_{\Omega}-u\rho_{j}(x)\eta'_{\tau}(t)dxdt = -\int_{\Omega}u(x, t_{1})\rho_{j}(x)dx +\int_{\Omega}u(x, t_{2})\rho_{j}(x)dx. \end{equation*} | (6.5) |
We assume that a sequence \{\rho_{j}(x)\} of test functions in \Omega such that
\begin{equation*} \rho\in C^{2}(\Omega)\;\;, \;\;\rho_{j}(x)\rightarrow \rho(x)\quad \text{with}\quad\rho(x)\in C^{2}(\Omega) \end{equation*} |
and
\begin{equation*} \;\; \;\;\Delta\rho_{j}(x)\rightarrow \Delta\rho(x)\quad\text{uniformly in }\;\;\Omega. \end{equation*} |
Then for every t\in(0, T)\setminus D_{j} , there holds
\begin{equation*} \int_{\Omega}u(x, t)\rho_{j}(x)dx -\int_{\Omega} u_{0}\rho_{j}(x)dx = \int_{0}^{t}\int_{\Omega}\psi(u_{r})\Delta\rho_{j}(x)dxds+ \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{\partial\Omega}g(u_{r})\rho_{j}(x)d\mathcal{H}(x)dt+ \int_{0}^{t}\int_{\Omega}h(t)f\rho_{j}(x)dxdt. \end{equation*} | (6.6) |
By Dominated Convergence Theorem, we obtain
\begin{equation*} \int_{\Omega}u(x, t)\rho(x)dx -\int_{\Omega}u_{0}\rho(x)dx = \int_{0}^{t}\int_{\Omega}\psi(u_{r})\Delta\rho(x)dxds+ \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{\partial\Omega}g(u_{r})\rho(x)d\mathcal{H}(x)dt+ \int_{0}^{t}\int_{\Omega}h(t)f\rho(x)dxdt \end{equation*} | (6.7) |
for every t\in(0, T)\setminus D with D = \bigcup_{j\geq0}D_{j}
Since \psi(u_{r})\in L^{\infty}(Q) , for every \rho\in C^{2}(\Omega) and for every sequence \{t_{j}\}\subseteq(0, T)\setminus D , t_{j}\rightarrow 0^{+} as j\rightarrow \infty such that
\begin{equation*} \int_{\Omega}u(x, t_{j})\rho(x)dx -\int_{\Omega}u_{0}\rho(x)dx = \int_{0}^{t_{j}}\int_{\Omega}\psi(u_{r})\Delta\rho(x)dxds+ \end{equation*} |
\begin{equation*} +\int_{0}^{t_{j}}\int_{\partial\Omega}g(u_{r})\rho(x)d\mathcal{H}(x)dt+ \int_{0}^{t_{j}}\int_{\Omega}h(t)f\rho(x)dxdt \end{equation*} | (6.8) |
holds true.
Since u\in L^{\infty}((0, T), \mathcal{M}^{+}(\Omega)) , then we have
\begin{equation*} \sup\limits_{j}\parallel u(\cdot, t_{j})\parallel_{\mathcal{M}^{+}(\Omega)}\leq C. \end{equation*} | (6.9) |
So that there exists a subsequence \{t_{j_{m}}\}\subseteq \{t_{j}\} and a Radon measure \mu_{0}\in \mathcal{M}^{+}(\Omega) such that
\begin{equation*} u(\cdot, t_{j_{m}})\overset{*}\rightharpoonup \mu_{0}\;\;\;\text{in}\;\; \mathcal{M}^{+}(\Omega)\;\;\; \text{as}\;\;\; j_{m}\rightarrow \infty. \end{equation*} | (6.10) |
By the standard density arguments, one has
\begin{equation*} \text{ess}\lim\limits_{j_{m}\rightarrow \infty}\langle u(\cdot, t_{j_{m}}), \rho\rangle_{\Omega} = \langle u_{0}, \rho\rangle_{\Omega} \end{equation*} | (6.11) |
where \mu_{0} = u_{0} , hence (6.1) is obtained.
Proof of Theorem 2.4 Let u_{1}\;, \; u_{2} be two very weak solutions obtained as limit of approximation of (P) with initial data u_{01n} and u_{02n} respectively. Let \left\{u_{1n}\right\} , \left\{u_{2n}\right\}\subseteq L^{\infty}(Q)\cap L^{2}((0, T), H^{1}(\Omega)) be two approximating sequence solutions to the problem (P_{n}) . We consider a test function \xi\in C^{2, 1}(Q) such that \xi(\cdot, T) = 0 in \Omega and \frac{\partial\xi}{\partial\eta} = 0 on \partial\Omega\times(0, T) in the approximation problem (P_{n}) in the sense of the Definition 2.3, then there holds
\begin{equation*} \int_{Q}\left(u_{1n}-u_{2n}\right)\xi_{t}dxdt = -\int_{Q}\left(\psi_{n} (u_{1n})-\psi_{n} (u_{2n})\right)\Delta\xi dxdt- \end{equation*} |
\begin{equation*} -\int_{Q}h(t)\left(f_{1n}-f_{2n}\right)\xi dxdt-\int_{S}\left(g_{1n}-g_{2n}\right)\xi d\mathcal{H}(x)dt- \end{equation*} |
\begin{equation*} \int_{\Omega}\left(u_{01n}-u_{02n}\right)\xi(x, 0)dx, \end{equation*} | (6.12) |
where \left\{f_{1n}\right\} , \left\{f_{2n}\right\} , \left\{u_{01n}\right\} , and \left\{u_{02n}\right\} are two approximating functions.
By the assumption g(u_{r}) = L a.e in S , then for any sequences \left\{u_{1n}\right\} , \left\{u_{2n}\right\} one has g(u_{1n} = g(u_{2n}) = L on S . Consequently the third term on the right hand-side of the equation (6.12) vanishes.
For almost every (x, t)\in Q , we consider the function a_{n}(x, t) defined as
\begin{equation*} a_{n}(x, t) = \begin{cases} \frac{\psi_{n}(u_{1n}(x, t))-\psi_{n}(u_{2n}(x, t))}{u_{1n}(x, t)-u_{2n}(x, t)}\quad&\text{if}\, \, \, u_{1n}(x, t)\neq u_{2n}(x, t), \\ \psi_{n}'(u_{1n}(x, t))\quad&\text{if}\, \, \, u_{1n}(x, t) = u_{2n}(x, t). \end{cases} \end{equation*} | (6.13) |
Obviously a_{n}\in L^{\infty}(Q) and for every n\in \mathbb{N} there exists a positive constant C_{n} such that
\begin{equation*} \text{ess}\inf\limits_{(x, t)\in Q}a_{n}(x, t)\geq C_{n} > 0. \end{equation*} | (6.14) |
This ensures that for every z\in C^{2}_{c}(Q) , the problem
\begin{equation*} \begin{cases} \xi_{nt}+a_{n}\Delta\xi_{n}+z = 0\quad&\text{in}\, \, \, Q, \\ \frac{\partial\xi_{n}}{\partial\eta} = 0\quad&\text{on}\, \, \, S, \\ \xi_{n}(\cdot, T) = 0 \quad&\text{in}\, \, \, \Omega, \end{cases} \end{equation*} | (6.15) |
has a unique solution \xi_{n}\in L^{\infty}((0, T), H^{2}(\Omega))\cap L^{2}((0, T), H^{1}(\Omega)) with \xi_{nt}\in L^{2}(Q) (see [18,21]).
Moreover, it can be seen that
\begin{equation*} \mid\xi_{n}(x, t)\mid\leq(T-t)\parallel z\parallel_{L^{\infty}(Q)}. \end{equation*} | (6.16) |
Let us consider the function \beta such that for any t_{1}+1 < t_{2} and t_{1}\;, \; t_{2}\in (0, T)
\begin{equation*} \beta(t) = \begin{cases} 0\quad&\text{if}\, \, \, 0\leq t\leq t_{1}, \\ t-t_{1}\quad&\text{if}\, \, \, t_{1} < t < t_{2}, \\ t_{2}-t_{1}\quad&\text{if}\, \, \, t\geq t_{2}. \end{cases} \end{equation*} | (6.17) |
Choosing \beta\Delta\xi_{n} as a test function in (6.15), then we obtain
\begin{equation*} \int_{Q}{\xi_{nt}\beta(t)\Delta\xi_{n}}dxdt+\int_{Q}\beta(t)a_{n}(x, t)[\Delta\xi_{n}]^{2}dxdt +\int_{Q}z\beta(t)\Delta\xi_{n}dxdt = 0. \end{equation*} | (6.18) |
It follows that
\begin{equation*} \frac{1}{2}\int_{Q}\mid\nabla \xi_{n}\mid^{2}dxdt+\int_{Q}a_{n}(x, t)[\Delta\xi_{n}]^{2}dxdt\leq C_{0}(T, z) \end{equation*} | (6.19) |
holds, for some constant C_{0}(T, z) independent on n .
From (6.16) and (6.19), there exists a constant C_{1}(T, z) such that
\begin{equation*} \parallel\xi_{n}\parallel_{L^{2}((0, T), H^{1}(\Omega))}+\parallel \sqrt{a_{n}}\Delta\xi_{n}\parallel_{L^{2}(Q)}\leq C_{1}(T, z). \end{equation*} | (6.20) |
On the other hand, multiplying (6.15) by \Delta\xi_{n} and we obtain
-\int_{Q}\nabla\xi_{n}\nabla\xi_{nt}+\int_{Q}a_{n}[\Delta\xi_{n}]^{2}dxdt = -\int_{Q}\xi_{n}\Delta z dxdt |
which leads to
\begin{equation*} \frac{1}{2}\int_{\Omega}\mid\nabla \xi_{n}\mid^{2}(x, 0)dx+\int_{Q}a_{n}[\Delta\xi_{n}]^{2}dxdt\leq C_{2}(T, z), \end{equation*} | (6.21) |
where C_{2}(T, z) = C\left(\parallel\xi_{n}\parallel_{L^{\infty}(Q)}, \parallel z\parallel_{C^{2}(\overline{Q})}\right) > 0 . Therefore, we get
\begin{equation*} \parallel\xi_{n}(\cdot, 0)\parallel_{H^{1}(\Omega)}+\parallel \sqrt{a_{n}}\Delta\xi_{n}\parallel_{L^{2}(Q)}\leq C_{2}(T, z). \end{equation*} | (6.22) |
By standard density arguments, we can choose \xi = \xi_{n} as a test function in (6.15). It implies that (6.12) yields
\begin{equation*} \int_{Q}\left(u_{1n}-u_{2n}\right)zdxdt = \int_{Q}h(t)\left(f_{1n}-f_{2n}\right)\xi_{n}(x, t)dxdt+ \end{equation*} |
\begin{equation*} +\int_{\Omega}\left(u_{01n}-u_{02n}\right)\xi_{n}(x, 0) dx. \end{equation*} | (6.23) |
Letting n to infinity in (6.23). Then it enough to observe from (6.20), there exists \xi_{n}\in L^{\infty}((0, T), H^{2}(\Omega))\cap L^{2}((0, T), H^{1}(\Omega)) which is obtained by extracting the subsequence of the \left\{\xi_{n}\right\} such that
\begin{equation*} \xi_{n}(x, t)\overset{*}\rightharpoonup \xi(x, t)\, \, \, \text{in}\, \, \, \, L^{\infty}(Q). \end{equation*} | (6.24) |
\begin{equation*} \xi_{n}(x, t)\rightharpoonup \xi(x, t)\, \, \, \text{in}\, \, \, \, L^{2}((0, T), H^{1}(\Omega)). \end{equation*} | (6.25) |
Since \xi_{nt}\in L^{2}(Q) and the compactness theorem states in [21], we deduce that
\begin{equation*} \xi_{nt}(x, t)\rightarrow \xi_{t}(x, t)\, \, \, \text{in}\, \, \, \, L^{2}((0, T), (H^{1}(\Omega))^{*}), \end{equation*} | (6.26) |
\begin{equation*} \xi_{n}(x, t)\rightarrow \xi(x, t)\, \, \, \text{a.e in}\, \, \, \, Q. \end{equation*} | (6.27) |
By (6.16) and (6.22), there exists \xi(\cdot, 0)\in L^{\infty}(\Omega)\cap H^{1}(\Omega) such that the following statements
\begin{equation*} \xi_{n}(x, 0)\overset{*}\rightharpoonup \xi(x, 0)\, \, \, \text{in}\, \, \, \, L^{\infty}(\Omega), \end{equation*} | (6.28) |
\begin{equation*} \xi_{n}(x, 0)\rightharpoonup\xi(x, 0)\, \, \, \text{in}\, \, \, \, H^{1}(\Omega) \end{equation*} | (6.29) |
holds true. By Theorem 2.1, the solutions of the problem (H) are unique in \mathcal{M}^{+}(Q) . Therefore f_{1n}\overset{*}\rightharpoonup f in \mathcal{M}^{+}(Q) and f_{2n}\overset{*}\rightharpoonup f in \mathcal{M}^{+}(Q) . Furthermore, the sequences \{u_{01n}\} and \{u_{02n}\} satisfy the assumption (2.6). By combining the above assumptions and Dominated Convergence Theorem, the Eq (6.23) reads
\begin{equation*} \int_{Q}(u_{1}-u_{2})z(x, t)dxdt = \lim\limits_{n\rightarrow +\infty}\int_{Q}\left[h(t)(f_{1n}-f_{2n})\right]\xi(x, t)dxdt+ \end{equation*} |
\begin{equation*} +\lim\limits_{n\rightarrow \infty}\int_{\Omega}\left( f_{01n}-f_{02n}\right)\xi(x, 0)dxdt -\lim\limits_{n\rightarrow \infty}\int_{\Omega}\left( F_{01n}-F_{02n}\right)\xi(x, 0)dx = 0 \end{equation*} |
It follows that u_{1} = u_{2} in \mathcal{M}^{+}(Q) .
In this section, we prove the result of decay estimate solutions.
Proof of Theorem 2.5. We consider u_{n} and v_{n} two solutions of the approximation problems (P_{n}) and (P_{0n}) respectively. For any \xi\in C^{1}((0, T), C^{1}(\Omega)) such that \xi(\cdot, T) = 0 in \Omega and \frac{\partial\xi}{\partial\eta} = 0 on S as a test function of the approximation problem (P_{n})-(P_{0n}) , then there holds
\begin{equation*} \int_{Q}(u_{n}-v_{n})\xi_{t}(x, t)dxdt = \int_{Q}\nabla[\psi(u_{n})-\vartheta(v_{n})]\nabla\xi dxdt-\int_{Q}h(t)f_{n}(x, t)\xi dxdt- \end{equation*} |
\begin{equation*} -\int_{S}(g(u_{n})-g_{1}(v_{n}))\xi d\mathcal{H}(x)dt. \end{equation*} | (7.1) |
For every \epsilon > 0 , we consider \{z_{\epsilon}\} be a sequence of smooth functions such that \parallel z_{\epsilon} \parallel_{L^{1}(0, T)}\leq C and z_{\epsilon}(t)\overset{*}\rightharpoonup\delta_{t} in \mathcal{M}^{+}(0, T) . Let us choose \xi(x, t) = \text{sign}(u_{n}(x, t)-v_{n}(x, t))\int^{T}_{t}z_{\epsilon}(s)(T-s)^{\alpha}ds(\alpha > 1) into the Eq (7.1), then (7.1) reads
\begin{equation*} \left[\int_{0}^{T}z_{\epsilon}(t)(T-t)^{\alpha}dt\right]\left[\int_{\Omega}\mid u_{n}(\cdot, t)-v_{n}(\cdot, t)\mid dx\right] \end{equation*} |
\begin{equation*} = -\left[\int_{t}^{T}h(t)\left(\int_{0}^{T}z_{\epsilon}(s)(T-s)^{\alpha}ds\right)dt\right]\left[\int_{\Omega} f_{n}\text{sign}(u_{n}(x, t)-v_{n}(x, t))dx\right]- \end{equation*} |
\begin{equation*} -\left[\int_{t}^{T}\left(\int_{0}^{T}z_{\epsilon}(s)(T-s)^{\alpha}ds\right)dt\right]\left[\int_{\partial\Omega} (g(u_{n})-g_{1}(u_{n}))\text{sign}(u_{n}(x, t)-v_{n}(x, t))d\mathcal{H}(x)\right] \end{equation*} | (7.2) |
Letting \epsilon\rightarrow 0^{+} in the previous equation and using the properties of the Dirac mass at t , then we have the following expression
\begin{equation*} (T-t)^{\alpha}\int_{\Omega}\mid u_{n}(\cdot, t)-v_{n}(\cdot, t)\mid dx\leq C \int_{Q} f_{n}(x, t) (T-t)^{\alpha}dxdt \end{equation*} | (7.3) |
for any t\in (0, T)\backslash H^{*} with \mid H^{*}\mid = 0 and C = C(\mid S\mid, \parallel g(u_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}, \parallel g_{1}(v_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}, T^{\alpha}) > 0 is a constant. On the other hand, by (4.5) we have
\begin{equation*} f_{n}(x, t)(T-t)^{\alpha} = T^{\alpha}\int_{\Omega}G_{N}(x-y, t)u_{0n}(y)dy+ \int_{0}^{t}\int_{\partial\Omega}G_{N}(x-y, t-\sigma) g(f_{n})(T-\sigma)^{\alpha}dyd\sigma+ \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{\Omega}G_{N}(x-y, t-\sigma)\left\{-\alpha f_{n}(T-\sigma)^{\alpha-1}+\mu_{n}(T-\sigma)^{\alpha}\right\}dyd\sigma. \end{equation*} |
By (2.11)-(2.13) and the properties of the Green function G_{N} , we get the following result
\begin{equation*} \int_{Q} f_{n}(x, t)(T-t)^{\alpha}dxdt\leq T^{\alpha+1}\int_{\Omega}u_{0n}(y)dy+\alpha\int_{0}^{t}\int_{Q} f_{n}(y, \sigma) (T-\sigma)^{\alpha}dyd\sigma dt+ \end{equation*} |
\begin{equation*} +\int_{0}^{t}\int_{Q} \mu_{n} (T-\sigma)^{\alpha}dyd\sigma dt+\int_{0}^{T}\int_{S} g(f_{n})(T-\sigma)^{\alpha}dyd\sigma dt. \end{equation*} |
By the assumptions (A), (4.1) and (4.2), there exists a positive constant C = C(T^{\alpha+1}, \parallel g(f_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}, \mid S\mid) > 0 such that
\begin{equation*} \int_{Q} f_{n}(x, t)(T-t)^{\alpha}dxdt\leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(\Omega)})+ \alpha\int_{0}^{t}\left(\int_{Q} f_{n}(x, \sigma) (T-\sigma)^{\alpha}dxdt\right)d\sigma. \end{equation*} | (7.4) |
By Gronwall's inequality, (7.4) yields
\begin{equation*} \int_{Q} f_{n}(x, t)(T-t)^{\alpha}dxdt\leq Ce^{\alpha T}(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(\Omega)}) \end{equation*} | (7.5) |
where C = C(T^{\alpha+1}, \parallel g(f_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}, \mid S\mid, e^{\alpha T}) > 0 is a constant. Combining (7.3) with (7.5), we deduce that
\begin{equation*} (T-t)^{\alpha}\int_{\Omega}\mid u_{n}(\cdot, t)-v_{n}(\cdot, t)\mid dx\leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}). \end{equation*} | (7.6) |
By [24, Chapter V, Section 5.2.1, Theorem 1], the semi-continuity of the total variation yields,
\begin{equation*} (T-t)^{\alpha}\parallel u(\cdot, t)-v(\cdot, t) \parallel_{\mathcal{M}^{+}(\Omega)}\leq (T-t)^{\alpha}\liminf\limits_{n\rightarrow \infty}\int_{\Omega}\mid u_{n}(\cdot, t)-v_{n}(\cdot, t)\mid dx \end{equation*} |
\begin{equation*} \;\;\qquad\qquad\qquad\qquad\quad\leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}). \end{equation*} | (7.7) |
Hence (2.18) holds.
We consider u_{n} and w_{n} two solutions of the approximation problems (P_{n}) and (P_{1n}) respectively. For any \xi\in C^{1}((0, T), C^{1}(\Omega)) such that \xi(\cdot, T) = 0 in \Omega and \frac{\partial\xi}{\partial\eta} = 0 on S as a test function of the approximation problem (P_{n})-(P_{1n}) . Therefore, we have the following equation
\begin{equation*} \int_{Q}(u_{n}-w_{n})\xi_{t}(x, t)dxdt = \int_{Q}\nabla[\psi(u_{n})-\psi(w_{n})]\nabla\xi dxdt-\int_{\Omega}u_{0n}\xi(x, 0) dxdt \end{equation*} | (7.8) |
Taking \xi(x, t) = \text{sign}(u_{n}(x, t)-w_{n}(x, t))\int^{T}_{t}z_{\epsilon}(s)(T-s)^{\alpha}ds(\alpha > 1) into the equality (7.8), then we obtain
\begin{equation*} \left[\int_{0}^{T}z_{\epsilon}(t)(T-t)^{\alpha}dt\right]\left[\int_{\Omega}\mid u_{n}(\cdot, t)-w_{n}(\cdot, t)\mid dx\right] \end{equation*} |
\begin{equation*} = -\left[\int_{0}^{T}z_{\epsilon}(s)(T-s)^{\alpha}dt\right]\left[\int_{\Omega} u_{0n}\text{sign}(u_{n}(x, 0)-w_{n}(x, 0))\right]. \end{equation*} |
Letting \epsilon\rightarrow 0^{+} in the previous equation and using the properties of the Dirac mass at t , then we have
\begin{equation*} (T-t)^{\alpha}\int_{\Omega}\mid u_{n}(\cdot, t)-w_{n}(\cdot, t)\mid dx\leq T^{\alpha}\int_{\Omega} u_{0n} dx. \end{equation*} | (7.9) |
By (4.1), the above inequality (7.9) yields
\begin{equation*} (T-t)^{\alpha}\int_{\Omega}\mid u_{n}(\cdot, t)-w_{n}(\cdot, t)\mid dx\leq C\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}. \end{equation*} | (7.10) |
By [24, Chapter V, Section 5.2.1, Theorem 1], the semi-continuity of the total variation yields,
\begin{equation*} (T-t)^{\alpha}\parallel u(\cdot, t)-w(\cdot, t) \parallel_{\mathcal{M}^{+}(\Omega)}\leq (T-t)^{\alpha}\liminf\limits_{n\rightarrow \infty}\int_{\Omega}\mid u_{n}(\cdot, t)-w_{n}(\cdot, t)\mid dx \end{equation*} |
\begin{equation*} \;\;\qquad\leq C\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)} \end{equation*} |
where C = C(T, \alpha) > 0 a constant. Hence (2.19) is achieved. Now we consider the auxiliary function \mathcal{W}_{n} such that
\begin{equation*} \mathcal{W}_{n}(x, t) = t^{\alpha}u_{n}(x, t)\text{sign}(\mathcal{W}_{n}) \end{equation*} | (7.11) |
for every \alpha > 1 . The derivation of the expression \mathcal{W}_{n} with respect to the variable t gives
\begin{equation*} \mathcal{W}_{nt}(x, t) = \alpha t^{\alpha-1}u_{n}(x, t)\text{sign}(\mathcal{W}_{n})+ t^{\alpha}u_{nt}(x, t)\text{sign}(\mathcal{W}_{n}). \end{equation*} | (7.12) |
Since u_{nt} = \Delta\psi(u_{n})+h(t)f_{n}(x, t) and we multiply the Eq (7.12) by the function \text{sign}(\mathcal{W}_{n}) and then we integrate the result over \Omega\times(0, t) (for any t\in(0, T) ), then we obtain
\begin{equation*} \int_{\Omega}\mid \mathcal{W}_{n}\mid(x, t) dx = \alpha\int_{0}^{t}s^{\alpha-1}\int_{\Omega}u_{n}(x, s)dxds+ \int_{0}^{t}\int_{\partial\Omega}g(u_{n})s^{\alpha}d\mathcal{H}(x)ds+\int_{0}^{t} \int_{\Omega}s^{\alpha}h(s)f_{n}(x, s)dxds. \end{equation*} | (7.13) |
By replacing the expression of \mathcal{W}_{n} in (7.13), we deduce that
\begin{equation*} t^{\alpha}\int_{\Omega}u_{n}(x, t)dx\leq \alpha T^{\alpha}\int_{Q}u_{n}(x, t)dxdt +T^{\alpha}\int_{S}g(u_{n})d\mathcal{H}(x)dt+h(T)T^{\alpha}\int_{Q}f_{n}(x, t)dxdt. \end{equation*} | (7.14) |
By assumptions (A) , (J) , (2.16) and (4.19), there exists a constant C = C(\alpha T^{\alpha+1}, h(T)T^{\alpha}, \parallel g(u_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}) > 0 such that
\begin{equation*} t^{\alpha}\int_{\Omega}u_{n}(x, t)dx\leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}). \end{equation*} | (7.15) |
According to [24, Chapter V, Section 5.2.1, Theorem 1], we conclude from the estimate (7.15), the following estimate
\begin{equation*} t^{\alpha}\parallel u(\cdot, t)\parallel_{\mathcal{M}^{+}(\Omega)}\leq t^{\alpha}\liminf\limits_{n\rightarrow \infty}\int_{\Omega} u_{n}(\cdot, t)dx \leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel \mu\parallel_{\mathcal{M}^{+}(Q)}). \end{equation*} |
Hence the estimate (2.21) is completed.
To show the existence of the problem (E) , we employ the natural approximation method. Therefore, the solution of the problem (P) is constructed by limiting point of a family \{u_{n}\} of solutions to the approximation problem. To this purpose, we consider the function \phi\in C^{\infty}_{c}(\Omega) such that 0\leq\phi\leq 1 and \phi = 1 in K_{0} (for any compact set K_{0}\subset\Omega\subset \mathbb{R}^{N} ), then we get
\begin{equation*} -\Delta(\phi\psi(U))+\phi U = \phi u_{0}+\varepsilon(\phi)\quad\text{in}\quad \mathcal{D}'(\Omega) \end{equation*} |
where \varepsilon(\phi) = -\psi(U)\Delta\phi-2\nabla\phi\nabla\psi(U) and \varepsilon(\phi) = 0 in K_{0} with \varepsilon(\phi)\in L^{1}(\Omega) .
Now we consider the approximation of problem (E)
\begin{equation*} \begin{cases} -\Delta\psi(U_{n})+U_{n} = u_{0n}\ \ &\text{in} \ \ \Omega, \\ \frac{\partial \psi(U_{n})}{\partial \eta} = g(U_{n}) \ \ &\text{on} \ \ \partial\Omega,\end{cases} \end{equation*} | (E_{n}) |
where u_{0n} = (\phi u_{0}+\varepsilon(\phi))\ast\rho_{n} and \{\rho_{n}\} a sequence of standard mollifiers. Furthermore, the sequence \{u_{0n}\}\subseteq C^{\infty}(\overline{\Omega}) satisfies the assumption (4.1).
Then for every n\in \mathbb{N} , there exists U_{n}\in H^{1}(\Omega)\cap L^{\infty}(\Omega) solution of the approximation problem (E_{n}) .
In the next Lemma, we state the technical estimates important for the proof of the existing solutions.
Lemma 8.1 Assume that (I) , (A) and u_{0}\in \mathcal{M}^{+}(\Omega) are satisfied. The sequence \{U_{n}\} be a weak solution of the approximation problem (E_{n}) . Then, there holds
\begin{equation*} \parallel U_{n}\parallel_{L^{1}(\Omega)}\leq C\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}, \end{equation*} | (8.1) |
\begin{equation*} \parallel\nabla\psi(U_{n}) \parallel_{L^{2}(\Omega)}+\parallel\psi(U_{n}) \parallel_{L^{2}(\Omega)}\leq C, \end{equation*} | (8.2) |
where C > 0 is a constant. Moreover, for every 1\leq p < \frac{N}{N-1} there holds
\begin{equation*} \parallel\nabla\psi(U_{n}) \parallel_{L^{p}(\Omega)}+\parallel\psi(U_{n}) \parallel_{L^{p}(\Omega)}\leq C, \end{equation*} | (8.3) |
where C = C(p) > 0 is a constant.
Proof of Lemma 8.1 We consider \varphi\in C^{1}(\overline{\Omega}) as a test function in the approximation problem (E_{n}) , then we have
\begin{equation*} \int_{\Omega}\nabla\psi(U_{n})\nabla\varphi dx+\int_{\Omega} U_{n}\varphi dx = \int_{\Omega} u_{0n}\varphi dx+\int_{\partial\Omega} g(U_{n})\varphi d\mathcal{H}(x) \end{equation*} | (8.4) |
Assume that \Omega_{-} = \left\{x\in\Omega/\; U_{n}(x)\leq 0\; \; \text{in the sense of}\; L^{1}(\Omega)\right\} and \varphi(x) = \inf_{x\in\Omega}\{U_{n}(x), 0\} . It is worth observing that \varphi(x)\in L^{1}(\Omega) . To show that U_{n}\geq 0 in \Omega , it is enough to prove that \varphi(x) = 0 in \Omega . Indeed, we choose \varphi(x) = \text{sign}(U_{n}(x)) , then we get
\begin{equation*} \int_{\Omega_{-}}\mid U_{n}(x) \mid dx = \int_{\Omega_{-}} u_{0n}(x)\text{sign}(U_{n}(x)) dx+ \int_{\partial\Omega_{-}}g(U_{n})\text{sign}(U_{n}(x))d\mathcal{H}(x)\leq0 \end{equation*} | (8.5) |
where u_{0n}\geq0 in \Omega and g > 0 in \mathbb{R}_{+} (see the assumption (A) ). Therefore \varphi(x) = 0 a.e in \Omega . Hence the solution of the approximation problem (E_{n}) , U_{n}(x)\geq0 a.e x\in\Omega .
Now we consider the regularizing sequence \{\mathcal{T}_{\epsilon}\}\subseteq C^{1}(\mathbb{R}_{+}) for every \epsilon > 0 such that
(i) 0\leq \mathcal{T}_{\epsilon}(s)\leq 1 in \mathbb{R}_{+} , \mathcal{T}_{\epsilon}(s) = 0 , \mathcal{T}'_{\epsilon}\geq0 in \mathbb{R}_{+} ,
(ii) \mathcal{T}_{\epsilon}(s)\rightarrow 1 as \epsilon\rightarrow0^{+} for every s\neq0 .
We choose \mathcal{T}_{\epsilon}(U_{n})\in H^{1}(\Omega)\cap L^{\infty}(\Omega) as a test function in the approximation problem (E_{n}) and by employing the assumptions (A) and (I) , then we get
\begin{equation*} \int_{\Omega}\mathcal{T}'_{\epsilon}(U_{n})\psi'(U_{n}(x))\mid\nabla U_{n}(x)\mid^{2} dx+\int_{\Omega}U_{n}(x) \mathcal{T}_{\epsilon}(U_{n})dx\leq C\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)} \end{equation*} | (8.6) |
where C = C(\parallel g(U_{n}) \parallel_{L^{\infty}(\mathbb{R}_{+})}, \mid\partial\Omega\mid) > 0 . Since \mathcal{T}'_{\epsilon}(U_{k})\psi'(U_{n}(x))\geq 0 in \mathbb{R}_{+} (see the hypothesis (I) ), then (8.6) reads
\begin{equation*} \int_{\Omega}U_{n}(x) \mathcal{T}_{\epsilon}(U_{n})dx\leq C \parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)} \end{equation*} | (8.7) |
Again, by considering the limit when \epsilon\rightarrow 0^{+} , the estimate (8.1) holds true. Now we consider another regularizing sequence \{\mathrm{T}_{\epsilon}\}\subseteq C^{1}(\mathbb{R}_{+}) for every \epsilon > 0 such that \mathrm{T}_{\epsilon}(s) = 1 if 0\leq s\leq \frac{1}{\epsilon} , \mathrm{T}_{\epsilon}(s) = \epsilon s if \frac{1}{\epsilon}\leq s\leq \frac{2}{\epsilon} , \mathrm{T}_{\epsilon}(s) = 2 if s\geq \frac{2}{\epsilon} . It is obvious to see that 1\leq \mathrm{T}_{\epsilon}(s)\leq 2 in \mathbb{R}_{+} . We take the function \varphi(s) = \int_{0}^{s}\mathrm{T}_{\epsilon}(\sigma)d\sigma and we choose \varphi(\psi(U_{n})) as a test function in (E_{n}) , then we obtain
\begin{equation*} \int_{\Omega}\mid \nabla\psi(U_{n})\mid^{2}\mathrm{T}_{\epsilon}(\psi(U_{n}))dx+\int_{\Omega}U_{n}\varphi(\psi(U_{n}))dx = \int_{\Omega}\varphi(\psi(U_{n}))u_{0n}dx+\int_{\partial\Omega}g(U_{n})\varphi(\psi(U_{n}))d\mathcal{H}(x). \end{equation*} | (8.8) |
Since 1\leq\mathrm{T}_{\epsilon}(\psi(U_{n}))\leq2 and \psi(U_{n})\leq\varphi(\psi(U_{n}))\leq2\psi(U_{n}) , therefore there exists a positive constant C such that
\begin{equation*} \int_{\Omega}\mid \nabla\psi(U_{n})\mid^{2}dx\leq C\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)} \end{equation*} | (8.9) |
where C = C(\parallel \psi(U_{n}) \parallel_{L^{\infty}(\mathbb{R}_{+})}, \parallel g(U_{n}) \parallel_{L^{\infty}(\mathbb{R}_{+})}, \mid\partial\Omega\mid) > 0 . By the assumption (I) , the statement \psi(U_{n})\in L^{2}(\Omega) holds. Whence the estimate (8.2) is achieved.
Again, recalling the Hölder's inequality, we get
\begin{equation*} \int_{\Omega}\mid\nabla\psi(U_{n})\mid^{p}dx\leq \left[\int_{\Omega}\frac{\mid\nabla\psi(U_{n})\mid^{2}}{(1+\psi(U_{n}))^{2}}dx\right]^{\frac{1}{q}} \left[\int_{\Omega}(1+\psi(U_{n}))^{q'}dx\right]^{\frac{1}{q'}} \end{equation*} |
where q: = \frac{2}{p} and q': = \frac{2}{2-p} . Therefore, there exists a positive constant C = C(p, \parallel \psi(U_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}) > 0 such that
\begin{equation*} \int_{\Omega}\mid\nabla\psi(U_{n})\mid^{p}dx\leq C \end{equation*} | (8.10) |
By the assumption (I) , the statement \psi(U_{n})\in L^{p}(\Omega) holds. Hence the estimate (8.3) is achieved.
Proof of Theorem 2.6. From the estimate (8.2) and assumption (A) , we can extract from \{\psi(U_{n})\} a subsequence \{\psi(U_{n_{k}})\} such that
\begin{equation*} \psi(U_{n_{k}})\rightarrow V\quad\text{in}\quad H^{1}(\Omega)\quad\text{and}\quad \psi(U_{n_{k}})\rightarrow V\quad\text{a.e in}\quad \Omega \end{equation*} | (8.11) |
\begin{equation*} g(U_{n_{k}})\overset{*}\rightharpoonup V \quad L^{\infty}(\partial\Omega)\quad\text{and}\quad g(U_{n_{k}})\rightarrow V\quad\text{a.e in}\quad \partial\Omega \end{equation*} | (8.12) |
By (8.3), the sequence \{\psi(U_{n_{k}})\}\subseteq BV(\Omega) and applying [44, Chapter IV, Section 1.1, Proposition 5], there exists a subsequence \{\psi(U_{n_{k}})\} and V_{1}\in \mathcal{M}^{+}(\Omega) such that the convergence
\begin{equation*} \psi(U_{n_{k}})\overset{*}\rightharpoonup V_{1} \quad\text{in}\;\; \mathcal{M}^{+}(\Omega). \end{equation*} | (8.13) |
By repeating the same method as in the Proposition 5.2, we deduce that
\begin{equation*} U_{n_{k}}\overset{*}\rightharpoonup U: = \psi^{-1}(V)+\lambda_{1}\;\;\text{in}\;\; \mathcal{M}^{+}(\Omega) \end{equation*} | (8.14) |
where U_{r} = \psi^{-1}(V) a.e in \Omega , U_{s} = \lambda_{1} in \mathcal{M}^{+}(\Omega) and U_{r} = g^{-1}(V) a.e in \partial\Omega .
By [24, Chapter V, Section 5.2.1, Theorem 1], the estimate (8.1) yields
\begin{equation*} \parallel U\parallel_{\mathcal{M}^{+}(\Omega)}\leq \liminf\limits_{n\rightarrow \infty}\int_{\Omega}U_{n}(x)dx\leq C\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}. \end{equation*} |
Hence the estimate (2.23) is completed.
Remark 8.1 The sets
\begin{equation*} \mathcal{S}_{0} = \left\{x\in \overline{\Omega}\setminus \psi(U_{r})(x) = \gamma\right\}\;\text{and}\;\mathcal{N}_{0} = \left\{x\in \overline{\Omega}\setminus g(U_{r})(x) = 0\right\} \end{equation*} |
have zero Lebesgue measure. Moreover \mathcal{S}_{0}\subseteq \mathcal{N}_{0} and supp (U_{s})\subseteq \mathcal{S}_{0} .
Proof of Theorem 2.7. We choose \xi(x, t) = \text{sign}(u_{n}(x, t)-U_{n}(x))\int^{T}_{t}z_{\epsilon}(s)s^{\alpha}ds(\alpha > 1) as a test function in the approximation problem (P_{n})-(E_{n}) , then we have
\begin{equation*} \int_{\Omega}\int_{0}^{T}\mid u_{n}(x, t)-U_{n}(x)\mid z_{\epsilon}(t)t^{\alpha}dt = \int_{\Omega}\int_{0}^{T}\mid u_{0n}(x)-U_{n}(x)\mid z_{\epsilon}(t)t^{\alpha}dtdx+ \end{equation*} |
\begin{equation*} +\int_{\Omega}\int_{0}^{T}[g(u_{n})-g(U_{n})] \text{sign}(u_{n}(x, t)-U_{n}(x))\int^{T}_{t}z_{\epsilon}(s)s^{\alpha}dsdtd\mathcal{H}(x)+ \end{equation*} |
\begin{equation*} +\int_{\Omega}\int_{0}^{T}u_{n}(x, t)-U_{n}(x)] \text{sign}(u_{n}(x, t)-U_{n}(x))\int^{T}_{t}z_{\epsilon}(s)s^{\alpha}dsdtdx+ \end{equation*} |
\begin{equation*} +\int_{\Omega}\int_{0}^{T}h(t)f_{n}(x, t) \text{sign}(u_{n}(x, t)-U_{n}(x))\int^{T}_{t}z_{\epsilon}(s)s^{\alpha}dsdtdx. \end{equation*} |
By the previous proof mentioned above, we deduce that
\begin{equation*} \int_{\Omega}\int_{0}^{T}\mid u_{n}(x, t)-U_{n}(x)\mid z_{\epsilon}(t)t^{\alpha}dt\leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel\mu\parallel_{\mathcal{M}^{+}(Q)}) \end{equation*} | (8.15) |
where C = C(\alpha T^{\alpha+1}, h(T), T^{\alpha}, \parallel g(u_{n})\parallel_{L^{\infty}(\mathbb{R}_{+})}, \mid S\mid) > 0 is a constant. By letting \epsilon\rightarrow 0^{+} , then (8.15) reads
\begin{equation*} t^{\alpha}\int_{\Omega}\mid u_{n}(x, t)-U_{n}(x)\mid dx\leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel\mu\parallel_{\mathcal{M}^{+}(Q)}). \end{equation*} | (8.16) |
By virtue of [24, Chapter V, Section 5.2.1, Theorem 1], then the semi-continuity of the total variation yields
\begin{equation*} t^{\alpha}\parallel u(\cdot, t)-U(\cdot)\parallel_{\mathcal{M}^{+}(\Omega)}\leq\liminf\limits_{n\rightarrow +\infty} t^{\alpha}\int_{\Omega}\mid u_{n}(x, t)-U_{n}(x)\mid dx \leq C(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel\mu\parallel_{\mathcal{M}^{+}(Q)}) \end{equation*} | (8.17) |
for almost every t\in(0, T) and \alpha > 1 . By considering to the limit as t\rightarrow +\infty in the following inequality
\begin{equation*} \parallel u(\cdot, t)-U(\cdot)\parallel_{\mathcal{M}^{+}(\Omega)}\leq \frac{C}{t^{\alpha}}(\parallel u_{0}\parallel_{\mathcal{M}^{+}(\Omega)}+\parallel\mu\parallel_{\mathcal{M}^{+}(Q)}). \end{equation*} |
Hence the statement (2.24) follows.
In this paper, we study the existence, uniqueness, decay estimates, and the asymptotic behavior of the Radon measure-valued solutions for a class of nonlinear parabolic equations with a source term and nonzero Neumann boundary conditions. To attain this, we use the natural approximation method, the definition of the weak solutions, and the properties of the Radon measure. Concerning the study of the existence and uniqueness of the solutions to the problem (P), we first show that the source term corresponding to the solution of the linear inhomogeneous heat equation with measure data is a unique Radon measure-valued. Moreover, we establish the decay estimates of these solutions by using the suitable test functions and the auxiliary functions. Finally, we analyze the asymptotic behavior of these solutions by establishing the decay estimate of the difference between the solution to the problem (P) and the solution to the steady state problem (E).
This work was partially supported by National Natural Sciences Foundation of China, grant No: 11571057
The authors declare no conflict of interest.
[1] | World Health Organization, Coronavirus disease (COVID-19), 2023. Available from: https://www.who.int/news-room/fact-sheets/detail/coronavirus-disease-(covid-19). |
[2] | COVID-19, 2020. Available from: https://www.cdc.gov/coronavirus/2019-ncov/your-health/about-covid-19.html. |
[3] | Tuberculosis (TB)-basic TB facts, 2019. Available from: https://www.cdc.gov/tb/topic/basics/default.html. |
[4] | World Health Organization, Tuberculosis and COVID- 19, 2020. Available from: https://www.who.int/teams/global-tuberculosis-programme/covid-19. |
[5] | H. Yang, S. Lu, COVID-19 and tuberculosis, J. Transl. Int. Med., 8 (2020), 59–65. https://doi.org/10.2478/jtim-2020-0010 |
[6] |
G. T. Mousquer, A. Peres, M. Fiegenbaum, Pathology of TB/COVID-19 co-infection: the phantom menace, Tuberculosis, 126 (2021), 102020. https://doi.org/10.1016/j.tube.2020.102020 doi: 10.1016/j.tube.2020.102020
![]() |
[7] |
B. Diao, C. Wang, Y. Tan, X. Chen, Y. Liu, L. Ning, et al., Reduction and functional exhaustion of T cells in patients with coronavirus disease 2019 (COVID-19), Front. Immunol., 11 (2020), 827. https://doi.org/10.3389/fimmu.2020.00827 doi: 10.3389/fimmu.2020.00827
![]() |
[8] |
M. Khayat, H. Fan, Y. Vali, COVID-19 promoting the development of active tuberculosis in a patient with latent tuberculosis infection: a case report, Respir. Med. Case Rep., 32 (2021), 101344. https://doi.org/10.1016/j.rmcr.2021.101344 doi: 10.1016/j.rmcr.2021.101344
![]() |
[9] |
M. Tadolini, L. R. Codecasa, J. M. García-García, F. X. Blanc, S. Borisov, J. W. Alffenaar, et al., Active tuberculosis, sequelae and COVID-19 co-infection: first cohort of 49 cases, Eur. Respir. J., 56 (2020), 2001398. https://doi.org/10.1183/13993003.01398-2020 doi: 10.1183/13993003.01398-2020
![]() |
[10] |
A. Abdoli, S. Falahi, A. Kenarkoohi, COVID-19-associated opportunistic infections: a snapshot on the current reports, Clin. Exp. Med., 22 (2022), 327–346. https://doi.org/10.1007/s10238-021-00751-7 doi: 10.1007/s10238-021-00751-7
![]() |
[11] |
M. Kretzschmar, Disease modeling for public health: added value, challenges, and institutional constraints, J. Public Health Policy, 41 (2020), 39–51. https://doi.org/10.1057/s41271-019-00206-0 doi: 10.1057/s41271-019-00206-0
![]() |
[12] |
Y. Wu, M. Huang, X. Wang, Y. Li, L. Jiang, Y. Yuan, The prevention and control of tuberculosis: an analysis based on a tuberculosis dynamic model derived from the cases of Americans, BMC Publ. Health, 20 (2020), 1173. https://doi.org/10.1186/s12889-020-09260-w doi: 10.1186/s12889-020-09260-w
![]() |
[13] |
L. N. Nkamba, T. T. Manga, F. Agouanet, M. L. Mann Manyombe, Mathematical model to assess vaccination and effective contact rate impact in the spread of tuberculosis, J. Biol. Dyn., 13 (2019), 26–42. https://doi.org/10.1080/17513758.2018.1563218 doi: 10.1080/17513758.2018.1563218
![]() |
[14] |
S. Liu, Y. Bi, Y. Liu, Modeling and dynamic analysis of tuberculosis in mainland China from 1998 to 2017: the effect of DOTS strategy and further control, Theor. Biol. Med. Modell., 2020 (2020), 17. https://doi.org/10.1186/s12976-020-00124-9 doi: 10.1186/s12976-020-00124-9
![]() |
[15] |
K. C. Chong, C. C. Leung, W. W. Yew, B. C. Y. Zee, G. C. H. Tam, M. H. Wang, et al., Mathematical modelling of the impact of treating latent tuberculosis infection in the elderly in a city with intermediate tuberculosis burden. Sci. Rep., 9 (2019), 4869. https://doi.org/10.1038/s41598-019-41256-4 doi: 10.1038/s41598-019-41256-4
![]() |
[16] |
T. A. Perkins, G. Espa\tilde{n}a, Optimal control of the COVID-19 pandemic with non-pharmaceutical interventions, Bull. Math. Biol., 82 (2020), 118. https://doi.org/10.1007/s11538-020-00795-y doi: 10.1007/s11538-020-00795-y
![]() |
[17] |
S. I. Oke, M. I. Ekum, O. J. Akintande, M. O. Adeniyi, T. A. Adekiya, O. J. Achadu, et al., Optimal control of the coronavirus pandemic with both pharmaceutical and non-pharmaceutical interventions, Int. J. Dyn. Control, 11 (2023), 2295–2319. https://doi.org/10.1007/s40435-022-01112-2 doi: 10.1007/s40435-022-01112-2
![]() |
[18] |
M. Zamir, F. Nadeem, M. A. Alqudah, T. Abdeljawad, Future implications of COVID-19 through mathematical modeling, Results Phys., 33 (2022), 105097. https://doi.org/10.1016/j.rinp.2021.105097 doi: 10.1016/j.rinp.2021.105097
![]() |
[19] |
L. Masandawa, S. S. Mirau, I. S. Mbalawata, Mathematical modeling of COVID-19 transmission dynamics between healthcare workers and community, Results Phys., 29 (2021), 104731. https://doi.org/10.1016/j.rinp.2021.104731 doi: 10.1016/j.rinp.2021.104731
![]() |
[20] |
A. O. Atede, A. Omame, S. C. Inyama, A fractional order vaccination model for COVID-19 incorporating environmental transmission: a case study using Nigerian data, Bull. Biomath., 1 (2023), 78–110. https://doi.org/10.59292/bulletinbiomath.2023005 doi: 10.59292/bulletinbiomath.2023005
![]() |
[21] |
B. Yang, Z. Yu, Y. Cai, The impact of vaccination on the spread of COVID-19: studying by a mathematical model, Phys. A, 590 (2022), 12671. https://doi.org/10.1016/j.physa.2021.126717 doi: 10.1016/j.physa.2021.126717
![]() |
[22] |
A. Kouidere, O. Balatif, M. Rachik, Cost-effectiveness of a mathematical modeling with optimal control approach of spread of COVID-19 pandemic: a case study in Peru, Chaos Solitons Fract., 10 (2023), 100090. https://doi.org/10.1016/j.csfx.2022.100090 doi: 10.1016/j.csfx.2022.100090
![]() |
[23] |
C. N. Ngonghala, E. Iboi, S. Eikenberry, M. Scotch, C. R. MacIntyre, M. H. Bonds, et al., Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel Coronavirus, Math. Biosci., 325 (2020), 108364. https://doi.org/10.1016/j.mbs.2020.108364 doi: 10.1016/j.mbs.2020.108364
![]() |
[24] |
O. Sharomi, C. N. Podder, A. B. Gumel, B. Song, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Math. Biosci. Eng., 5 (2008), 145–174. https://doi.org/10.3934/mbe.2008.5.145 doi: 10.3934/mbe.2008.5.145
![]() |
[25] |
Z. Mukandavire, A. B. Gumel, W. Garira, J. M. Tchuenche, Mathematical analysis of a model for HIV-malaria co-infection, Math. Biosci. Eng., 6 (2009), 333–362. https://doi.org/10.3934/mbe.2009.6.333 doi: 10.3934/mbe.2009.6.333
![]() |
[26] |
H. T. Alemneh, A co-infection model of dengue and leptospirosis diseases, Adv. Differ. Equations, 2020 (2020), 664. https://doi.org/10.1186/s13662-020-03126-6 doi: 10.1186/s13662-020-03126-6
![]() |
[27] |
I. M. Hezam, A. Foul, A. Alrasheedi, A dynamic optimal control model for COVID-19 and cholera co-infection in Yemen, Adv. Differ. Equations, 2021 (2021), 108. htttps://doi.org/10.1186/s13662-021-03271-6 doi: 10.1186/s13662-021-03271-6
![]() |
[28] |
V. Guseva, N. Doktorova, O. Krivorotko, O. Otpushchennikova, L. Parolina, I. Vasilyeva, et al., Building a seir-model for predicting the HIV/tuberculosis coinfection epidemic for russian territories with low TB burden, Int. J. Infect. Dis., 134 (2023), S4-S5. https://doi.org/10.1016/j.ijid.2023.05.028 doi: 10.1016/j.ijid.2023.05.028
![]() |
[29] |
A. Ahmad, M. Farman, A. Akgül, N. Bukhari, S. Imtiaz, Mathematical analysis and numerical simulation of co-infection of TB-HIV, Arab J. Basic Appl. Sci., 27 (2020), 431–441. https://doi.org/10.1080/25765299.2020.1840771 doi: 10.1080/25765299.2020.1840771
![]() |
[30] |
A. Omame, A. D. Okuonghae, U. E. Nwafor, B. U. Odionyenma, A co-infection model for HPV and syphilis with optimal control and cost-effectiveness analysis, Int. J. Biomath., 14 (2021), 2150050. https://doi.org/10.1142/S1793524521500509 doi: 10.1142/S1793524521500509
![]() |
[31] |
A. Omame, M. Abbas, C. P. Onyenegecha, A fractional-order model for COVID-19 and tuberculosis co-infection using Atangana-Baleanu derivative, Chaos Solitons Fract., 153 (2021), 111486. https://doi.org/10.1016/j.chaos.2021.111486 doi: 10.1016/j.chaos.2021.111486
![]() |
[32] |
A. Omame, M. Abbas, C. P. Onyenegecha, A fractional order model for the co-interaction of COVID-19 and Hepatitis B virus, Results Phys., 37 (2022), 105498. https://doi.org/10.1016/j.rinp.2022.105498 doi: 10.1016/j.rinp.2022.105498
![]() |
[33] |
H. Rwezaura, M. L. Diagne, A. Omame, A. L. de Espindola, J. M. Tchuenche, Mathematical modeling and optimal control of SARS-CoV-2 and tuberculosis co-infection: a case study of Indonesia, Model. Earth Syst. Environ., 8 (2022), 5493–5520. https://doi.org/10.1007/s40808-022-01430-6 doi: 10.1007/s40808-022-01430-6
![]() |
[34] |
R. I. Gweryina, C. E. Madubueze, V. P. Bajiya, F. E. Esla, Modeling and analysis of tuberculosis and pneumonia co-infection dynamics with cost-effective strategies, Results Control Optim., 10 (2023), 100210. https://doi.org/10.1016/j.rico.2023.100210 doi: 10.1016/j.rico.2023.100210
![]() |
[35] |
K. G. Mekonen, L. L. Obsu, Mathematical modeling and analysis for the co-infection of COVID-19 and tubercu-losis, Heliyon, 8 (2022), e11195. https://doi.org/10.1016/j.heliyon.2022.e11195 doi: 10.1016/j.heliyon.2022.e11195
![]() |
[36] |
K. G. Mekonen, S. F. Balcha, L. L. Obsu, A. Hassen, Mathematical modeling and analysis of TB and COVID-19 coinfection, J. Appl. Math., 2022 (2022), 2449710. https://doi.org/10.1155/2022/2449710 doi: 10.1155/2022/2449710
![]() |
[37] |
A. Selvam, S. Sabarinathan, B. V. S. Kumar, H. Byeon, K. Guedri, S. M. Eldin, et al., Ulam-Hyers stability of tuberculosis and COVID-19 co-infection model under Atangana-Baleanu fractal-fractional operator, Sci. Rep., 13 (2023), 9012. https://doi.org/10.1038/s41598-023-35624-4 doi: 10.1038/s41598-023-35624-4
![]() |
[38] |
S. R. Bandekar, M. Ghosh, A co-infection model on TB- COVID-19 with optimal control and sensitivity analysis, Math. Comput. Simul., 200 (2022), 1–31. https://doi.org/10.1016/j.matcom.2022.04.001 doi: 10.1016/j.matcom.2022.04.001
![]() |
[39] |
F. Inayaturohmat, N. Anggriani, A. K. Supriatna, A mathematical model of tuberculosis and COVID-19 coinfection with the effect of isolation and treatment, Front. Appl. Math. Stat., 8 (2022), 958081. htttps://doi.org/10.3389/fams.2022.958081 doi: 10.3389/fams.2022.958081
![]() |
[40] |
Z. S. Kifle, L. L. Obsu, Co-dynamics of COVID-19 and TB with COVID-19 vaccination and exogenous reinfection for TB: an optimal control application, Infect. Dis. Modell., 8 (2023), 574–602. https://doi.org/10.1016/j.idm.2023.05.005 doi: 10.1016/j.idm.2023.05.005
![]() |
[41] |
S. W. Teklu, Y. F. Abebaw, B. B. Terefe, D. K. Mamo, HIV/AIDS and TB co-infection deterministic model bifurcation and optimal control analysis, Inf. Med. Unlocked, 41 (2023), 101328. https://doi.org/10.1016/j.imu.2023.101328 doi: 10.1016/j.imu.2023.101328
![]() |
[42] |
B. S. Kotola, S. W. Teklu, Y. F. Abebaw, Bifurcation and optimal control analysis of HIV/AIDS and COVID-19 co-infection model with numerical simulation, PLoS One, 18 (2023), e0284759. https://doi.org/10.1371/journal.pone.0284759 doi: 10.1371/journal.pone.0284759
![]() |
[43] |
S. W. Teklu, Mathematical analysis of the transmission dynamics of COVID-19 infection in the presence of intervention strategies, J. Biol. Dyn., 16 (2022), 640–664. https://doi.org/10.1080/17513758.2022.2111469 doi: 10.1080/17513758.2022.2111469
![]() |
[44] |
S. W. Teklu, B. B. Terefe, D. K. Mamo, Y. F. Abebaw, Optimal control strategies on HIV/AIDS and pneumonia co-infection with mathematical modelling approach, J. Biol. Dyn., 18 (2024), 2288873. https://doi.org/10.1080/17513758.2023.2288873 doi: 10.1080/17513758.2023.2288873
![]() |
[45] |
D. Okuonghae, S. Omosigho, Analysis of a mathematical model for tuberculosis: what could be done to increase case detection, J. Theor. Biol., 269 (2011), 31–45. https://doi.org/10.1016/j.jtbi.2010.09.044 doi: 10.1016/j.jtbi.2010.09.044
![]() |
[46] |
S. M. Garba, J. M. S. Lubuma, B. Tsanou, Modeling the transmission dynamics of the COVID-19 pandemic in South Africa, Math. Biosci., 328 (2020), 108441. https://doi.org/10.1016/j.mbs.2020.108441 doi: 10.1016/j.mbs.2020.108441
![]() |
[47] |
S. A. Lauer, K. H. Grantz, Q. Bi, F. K. Jones, Q. Zheng, H. R. Meredith, et al., The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application, Ann. Int. Med., 172 (2020), 9. https://doi.org/10.7326/M20-0504 doi: 10.7326/M20-0504
![]() |
[48] |
A. Alemu, Z. W. Bitew, G. Seid, G. Diriba, E. Gashu, N. Berhe, et al., Tuberculosis in individuals who recovered from COVID-19: a systematic review of case reports, PLOS One, 17 (2022), 0277807. https://doi.org/10.1371/journal.pone.0277807 doi: 10.1371/journal.pone.0277807
![]() |
[49] |
D. Okuonghae, A. Omame, Analysis of a mathematical model for COVID-19 population dynamics in Lagos, Nigeria, Chaos Solitons Fract., 139 (2020), 110032. https://doi.org/10.1016/j.chaos.2020.110032 doi: 10.1016/j.chaos.2020.110032
![]() |
[50] | C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of R0 and its role in global stability, Inst. Math. Appl., 125 (2002), 229. |
[51] |
P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/s0025-5564(02)00108-6 doi: 10.1016/s0025-5564(02)00108-6
![]() |
[52] | J. P. LaSalle, The stability of dynamical systems, Society for Industrial and Applied Mathematics, 1976. |
[53] |
J. O. Akanni, A non-linear optimal control model for illicit drug use and terrorism dynamics in developing countries with time-dependent control variables, Decis. Anal. J., 8 (2023), 100281. https://doi.org/10.1016/j.dajour.2023.100281 doi: 10.1016/j.dajour.2023.100281
![]() |
[54] |
C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361–404. https://doi.org/10.3934/mbe.2004.1.361 doi: 10.3934/mbe.2004.1.361
![]() |
[55] |
A. Abidemi, J. O. Akanni, O. D. Makinde, A non-linear mathematical model for analysing the impact of COVID-19 disease on higher education in developing countries, Healthcare Anal., 3 (2023), 100193. https://doi.org/10.1016/j.health.2023.100193 doi: 10.1016/j.health.2023.100193
![]() |
[56] |
S. M. Blower, H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example, Int. Stat. Rev., 62 (1994), 229–243. https://doi.org/10.2307/1403510 doi: 10.2307/1403510
![]() |
[57] |
M. A. Sanchez, S. M. Blower, Uncertainty and sensitivity analysis of the basic reproductive rate: tuberculosis as an example, Amer. J. Epidemiol., 145 (1997), 1127–1137. https://doi.org/10.1093/oxfordjournals.aje.a009076 doi: 10.1093/oxfordjournals.aje.a009076
![]() |
[58] |
J. Wu, R. Dhingra, M. Gambhir, J. V. Remais, Sensitivity analysis of infectious disease models: methods, advances and their application, J. R. Soc. Interface, 10 (2013), 1018. https://doi.org/10.1098/rsif.2012.1018 doi: 10.1098/rsif.2012.1018
![]() |
[59] |
S. Olaniyi, J. O. Akanni, O. A. Adepoju, Optimal control and cost-effectiveness analysis of an illicit drug use population dynamics, J. Appl. Nonlinear Dyn., 12 (2023), 133–146. https://doi.org/10.5890/JAND.2023.03.010 doi: 10.5890/JAND.2023.03.010
![]() |
1. | Quincy Stévène Nkombo, Fengquan Li, Christian Tathy, Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations, 2021, 2021, 1687-1847, 10.1186/s13662-021-03668-3 |