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Research article Special Issues

Normalized solutions for the mixed dispersion nonlinear Schrödinger equations with four types of potentials and mass subcritical growth

  • Received: 12 December 2022 Revised: 15 March 2023 Accepted: 27 March 2023 Published: 28 April 2023
  • This paper is devoted to considering the attainability of minimizers of the L2-constraint variational problem

    mγ,a=inf{Jγ(u):uH2(RN),RN|u|2dx=a2},

    where

    Jγ(u)=γ2RN|Δu|2dx+12RN|u|2dx+12RNV(x)|u|2dx12σ+2RN|u|2σ+2dx,

    γ>0, a>0, σ(0,2N) with N2. Moreover, the function V:RN[0,+) is continuous and bounded. By using the variational methods, we can prove that, when V satisfies four different assumptions, mγ,a are all achieved.

    Citation: Cheng Ma. Normalized solutions for the mixed dispersion nonlinear Schrödinger equations with four types of potentials and mass subcritical growth[J]. Electronic Research Archive, 2023, 31(7): 3759-3775. doi: 10.3934/era.2023191

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  • This paper is devoted to considering the attainability of minimizers of the L2-constraint variational problem

    mγ,a=inf{Jγ(u):uH2(RN),RN|u|2dx=a2},

    where

    Jγ(u)=γ2RN|Δu|2dx+12RN|u|2dx+12RNV(x)|u|2dx12σ+2RN|u|2σ+2dx,

    γ>0, a>0, σ(0,2N) with N2. Moreover, the function V:RN[0,+) is continuous and bounded. By using the variational methods, we can prove that, when V satisfies four different assumptions, mγ,a are all achieved.



    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, ΩRN(N2) 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)ψL(R+)C2(R+),  ψ(0)=0,  ψ>0  in  R+,(ii)ψ,ψL(R+)  and  ψ(s)0ass+,(iii)ψ(s)γass+,(iv)ψψκinR+,for someκR+, (I)

    and

    {(i)gL(R+)C1(R+),  g<0  in  R+  and  g>0  in  R+,(ii)gL(R+)andg(s)0ass+, (A)

    where R+[0,+), R+(0,+) and γR+. 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[1e1(1+s)m]andg(s)=e1(1+s)m. (1.1)

    where 0<m1.

    The function h satisfies the following hypothesis

    hC1(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),fη=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

    T0ur(,t),ξt(,t)Ωdt+T0h(t)f(,t),ξ(,t)Ωdt+u0,ξ(,0)+
    +T0g(ur(,t)),ξΩdt=T0ψ(ur)(,t),ξ(,t)Ωdt (1.2)

    (see Definition 2.1). Here the measure u(,t) is defined for almost every t(0,T), urL1(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 N2 and 0u0L((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<tT and u0Wτ,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 u0L2(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 u0M+(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

    utdivA(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+1Q=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 u0M+(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 u0M+(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

    Ωψ(Ur)φdx+ΩUφdx=Ωφdu0(x)+Ωg(Ur)φdH(x)

    (see Definition 2.6), where UrL1(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,Uη+C(U)=g2  on  Ω, (A.5)

    where B(U)L1(Ω), C(U)L1(Ω), f2L1(Ω), g2L1(Ω) and the expression of the differential operator L in [34, Section 2]. The authors proved the existence, uniqueness and regularity of the solutions UW1,1(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 ¯μM(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)(1p<) 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 u0M+(Ω) and μM+(Q), a measure u is called a weak solution of problem (P), if uM+(Q) such that

    (i) uL((0,T),M+(Ω)),

    (ii) ψ(ur)L2((0,T),H1(Ω)),

    (iii) g(ur)L1(S),

    (iv) for every ξC1((0,T),C1(Ω)), ξ(,T)=0 in Ω, u satisfies the identity

    T0u(,t),ξt(,t)Ωdt+T0h(t)f(,t),ξ(,t)Ωdt+u0,ξ(,0)Ω+
    +T0g(ur(,t)),ξΩdt=T0ψ(ur),ξΩdxdt (2.1)

    where ur is the nonnegative density of the absolutely continuous part of Radon-measure with respect to the Lebesgue measure such that urL((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 ΩBj epigraph of a C1,1 function ζ:RN1R that is

    ΩBj={xBj/xN>ζ(¯x)}andΩBj={xBj/xN=ζ(¯x)}

    where x=(¯x,xN), the local coordinates with ¯x=(x1,x2,,xN1). We denote ϑ={¯x,xΩBj}RN1, the projection of ΩBj onto the (N1) first components, and ϑς={¯x,xsupp(ς)Ω}.

    If a function ϕ is defined on S, we denote ϕS the function defined on (BjQ)×[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 Fν on S in the sense that the following Gauss-green formula holds:

    (i) For all ξCc(¯Q),

    Tν,ξ=QξdivF+QξF

    where Tν,ξ depends only on ξS.

    (ii) If (Bj,ς,f) is an above subsequence localization near boundary, then for all ξCc([0,T)ׯΩ) there holds

    Tν,ξ=lims01s{Tsϑζ(¯x)+sζ(¯x)F(ζ(¯x)10)ξσdxNd¯xdt}lims01ss0ΩF(001)ξσdxdt (2.2)

    where the divergence of the fields,

    F(x,t)=(u(x,t)ψ(ur(x,t)))

    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 Ω is a strong C1,1 open subset of RN. Then there exists an unique trace Tη:W1,1(Ω)L1(Ω) such that

    Tη,ξ=Sg(ur)ξdH(x)dt (2.3)

    where the function g(ur)L1(S) and ξCc([0,T)ׯΩ).

    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 μM+(Q) and u0M+d,2(Ω), a measure u is called a very weak solution to problem (P) if uL((0,T),M+(Ω)) such that

    T0u(,t),ξt(,t)Ωdt+Qψ(ur)Δξdxdt+Qh(t)f(x,t)ξdxdt+Sg(u)ξdHdt+u0,ξ(0)Ω=0 (2.4)

    for every ξC2,1(¯Q), which vanishes on Ω×[0,T], for t=T.

    To prove the uniqueness of the problem (P) when f lies in M+(Q), we consider the following every weak solution gives below:

    Definition 2.4 Let u0M+d,2(Ω) and μM+(Q) such that

    u0=f0divG0,f0L1(Ω)andG0[L2(Ω)]N.

    A function u is called a very weak solutions obtained as limit of approximation, if

    unuinM+(Q) (2.5)

    where {un}L(Q)L2((0,T),H1(Ω)) is the sequences of weak solutions to problem (Pn) satisfies

    {u0n=f0nF0nCc(Ω),F0ndivG0in(H1(Ω)),f0nf0inL1(Ω). (2.6)

    We denote (H1(Ω)) the dual space of H1(Ω) and the embedding H1(Ω)L2(Ω)(H1(Ω)) holds.

    Definition 2.5 Let u0M+d,2(Ω) and μM+d,2(Q) such that

    u0=f0divG0,f0L1(Ω)andG0[L2(Ω)]N.
    μ=f1divG+φt,f1L1(Q),G[L2(Q)]NandφL2((0,T),H1(Ω)).

    A measure f is called a very weak solutions obtained as limit of approximation, if

    fnfinM+(Q) (2.7)

    where {un} and {fn}L(Q)L2((0,T),H1(Ω)) are the sequences of weak solutions to problem (Pn) and (Hn) respectively satisfy

    {μn=f1nFn+gntCc(Q),u0n=f0nF0nCc(Ω),f1nf1inL1(Q),FndivGinL2((0,T),(H1(Ω))),φnφinL2((0,T),H1(Ω)),F0ndivG0in(H1(Ω)),f0nf0inL1(Ω). (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

    unuinM+(Q),μnμinM+(Q)andu0nu0inM+(Ω).

    M+d,2(Ω) denotes the set of nonnegative measures on Ω which are diffuse with respect to the Newtonian capacity and the definition of the diffuse measure with respect to the parabolic capacity 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 u0M+(Ω) and μM+(Q) hold.

    (i) Then, there exists a nonnegative Radon measure-valued solution to the problem (H) in the space L((0,T),M+(Ω)) such that

    f(x,t)=ΩGN(xy,t)du0(y)+t0ΩGN(xy,tσ)dμ(y,σ)+t0ΩGN(xy,tσ)g(f(y,σ))dH(y)dσ (2.9)

    for almost every t(0,T). Furthermore, the Radon measure-valued solution f satisfies the following estimate

    f(,t)M+(Ω)eCt(μM+(Q)+u0M+(Ω)) (2.10)

    for any C=C(T) a positive constant.

    (ii) Suppose that u0M+d,2(Ω), μM+d,2(Q) and g(f)=¯K almost everywhere on S (¯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((0,T),M+(Ω)).

    We denote by GN(xy,ts) as the Green function of the heat equation under homogeneous Neumann boundary conditions. By [4], the Green function satisfies the following properties

    GN(xy,ts)0,x,yΩ,0s<t<T, (2.11)
    ΩGN(xy,ts)dx=1,yΩ,0s<t<T. (2.12)

    There exist two positive constants τ1 and τ2 such that

    |GN(xy,ts)1Ω|τ1eτ2(ts),x,yΩ,1+s<t. (2.13)
    limtsΩGN(xy,ts)ϕ(y)dy=ϕ(x) (2.14)

    for any ϕCc(Ω) and Ω is a Lebesgue measure of the set Ω.

    Remark 2.1 (i) For any test function ξC1((0,T),C1(Ω)) such that ξ(,T)=0 in Ω and ξη=0 on S, the inner product f(,t),ξ(,t)Ω in (2.1) is given by the following expression

    f(,t),ξ(,t)Ω=ΩΩG(xy,t)ξ(y,0)du0(y)dx+
    +Ωt0ΩGN(xy,tσ)(fξσ2fξfΔξ)dydσdx+
    +Ωt0ΩGN(xy,tσ)ξ(y,σ)dμ(y,σ)dx+
    +Ωt0ΩGN(xy,tσ)ξ(y,σ)g(f(y,σ))dH(y)dσdx (2.15)

    where ξσ is a first derivative order of ξ with respect σ.

    (ii) By the regularity properties of the Green function GN(xy,tσ) in [42], the solution of the problem (H) given by (2.9), fL2((0,T),H1(Ω)).

    (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 tΩGN(xy,tσ)h(σ)dμ(y,σ), tΩGN(xy,tσ)h(σ)g(f(y,σ))dH(y) and tΩGN(xy,tσ)f(y,σ)h(σ)dy are continuous in R+. Then there holds

    limt0+h(t)f(x,t)=limt0+t0ΩGN(xy,tσ)f(y,σ)h(σ)dydσ+
    +limt0+t0ΩGN(xy,tσ)h(σ)g(f)dH(y)dσ+limt0+t0ΩGN(xy,tσ)h(σ)dμ(y,σ)=0.

    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 μ and u0 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), μM+(Q) and u0M+(Ω) hold. For any function h satisfying (J), there exists t(0,T) such that t0h(σ)dσ=1 and u is a weak solution to the problem (P). Then μ and u0 are absolutely continuous measures with respect to the parabolic capacity.

    Notice that Newtonian and parabolic capacity are equivalent, then μ and u0 are absolutely continuous measures with respect to C2-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) μM+(Q) and u0M+(Ω) are satisfied. Then there exists a weak solution u to problem (P) obtained as a limiting point of the sequence {un} of solutions to problems (Pn) such that for every t(0,T)H, there holds

    u(,t)M+(Ω)C(μM+(Q)+u0M+(Ω)). (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), μM+d,2(Q) and u0M+d,2(Ω) hold. Then there exists a unique very weak solution obtained as the limit of approximation u of the problem (P), if g(ur)=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.

    {vt=Δϑ(v)  in  Q,ϑ(v)η=g1(v)  on  S,v(x,0)=u0  in  Ω, (P0)

    The functions ψ and g satisfy the assumption (I) and (A) respectively and have the same properties with the functions ϑ and g1 given as follows

    ϑ(s)=γ[11(1+s)m](m>0)andg1(s)=1(1+s)m (2.17)

    where m>0 and s>0. Therefore, by Theorem 2.3, the problem (P0) possesses a solution in the space L((0,T),M+(Ω)), such that

    v(,t)M+(Ω)Cu0M+(Ω)

    for almost every t(0,T).

    Similarly, we consider the following problem

    {wt=Δψ(w)+h(t)f(x,t)  in  Q,ψ(w)η=g(w)  on  S,w(x,0)=0  in  Ω, (P1)

    By Theorem 2.3, the problem (P1) admits a solution in L((0,T),M+(Ω)), such that

    w(,t)M+(Ω)CμM+(Ω)

    for almost every t(0,T).

    Now we state the decay estimates in the next theorem:

    Theorem 2.5 Suppose that (I), (J), (A), μM+(Q) and u0M+(Ω) 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 (P0) and w is the weak solution to the problem (P1). Then for every t(0,T)H with H∣=0, there holds

    u(,t)v(,t)M+(Ω)C(Tt)α(μM+(Q)+u0M+(Ω)), (2.18)
    u(,t)w(,t)M+(Ω)Cu0M+(Ω)(Tt)α, (2.19)

    and

    u(,t)M+(Ω)Ctα(u0M+(Ω)+μM+(Q)) (2.20)

    for any positive constant C and α>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,+), then we assume that the hypothesis

    lim supt+u(,t)M+(Ω)C (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 u0M+(Ω) are satisfied. A measure U is a solution of the problem (E), if UM+(Ω) such that

    (i) ψ(Ur)W1,1(Ω),

    (ii) g(Ur)L1(Ω),

    (iii) for every φC1(Ω), the following assertion

    Ωψ(Ur(x))φ(x)dx+ΩU(x)φ(x)dx=Ωφ(x)du0(x)+Ωg(U(x))φ(x)dH(x) (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 u0M+(Ω) are satisfied. Then there exists a weak solution UM+(Ω) of the problem (E) obtained as a limiting point of the sequence {Un} of solutions to the approximation problem (En) such that

    UM+(Ω)Cu0M+(Ω) (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), u0M+(Ω) and μM+(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

    u(,t)UinM+(Ω)ast (2.24)

    In the following section, we define the truncation function for k>0 and sR,

    Tk(s)=min{s,k}sign(s).

    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 EΩ, the C2-capacity of E in Ω is defined as

    C2(E)=inf{Ω(u2+u2)dx/uZEΩ}

    where ZEΩ denotes the set of u which belongs to H1(Ω) such that 0u1 almost everywhere in Ω, and u=1 almost everywhere in a neighborhood E.

    Let W={uL2((0,T),H1(Ω))andutL2((0,T),(H1(Ω)))} endowed with its natural norm uW=∥uL2((0,T),H1(Ω))+utL2((0,T),(H1(Ω))) a Banach space. For any open set UQ, we define the parabolic capacity as

    Cap(U)=inf{uW/uVUQ}

    where VUQ denotes the set of u belongs to W such that 0u1 almost everywhere in Q, and u=1 almost everywhere in a neighborhood U.

    Let M(B) be the space of bounded Radon measures on B, and M+(B)M(B) the cone of nonnegative bounded Radon measures on B. For any μM(B) a bounded Radon measure on B, we set

    μM(Ω):=∣μ(B)

    where μ stands for the total variation of μ.

    The duality map ,B between the space M(B) and Cc(B) is defined by

    μ,φB=Bφdμ.

    M+s(B) denotes the set of nonnegative measures singular with respect to the Lebesgue measure, namely

    M+s(B):={μM+(Ω)/ a Borel setFBsuch thatF∣=0,μ=μ

    we will consider either the Lebesgue measure on or . Similarly, the set of nonnegative measures absolutely continuous with respect to the Lebesgue measure, namely

    Let be the set of nonnegative measures on which are concentrated with respect to the Newtonian capacity

    denotes the set of nonnegative measures on which are diffuse with respect to the Newtonian capacity

    It is known that a measure (resp. ) if there exist and (resp. if , there exist , and ) such that

    (3.1)

    For any , if there exists a unique couple , such that

    (3.2)

    On the other hand, there exists a unique couple such that

    (3.3)

    where either , or , .

    By , the set of nonnegative Radon measures such that for every , there exists a measure such that

    (i) for every the map

    and

    (ii) there exists a constant such that

    with the norm denotes

    (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 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 , the approximation of problem such that

    ()

    Since , the approximation of the Radon measure is given by [9, Lemma 4.1] such that satisfies the following assumptions

    (4.1)

    Moreover , the approximation of the Radon measure is given by [15] such that fulfills the following hypotheses

    (4.2)

    for every . By [21,22,43], the approximation problem has a unique solution in .

    In the next proposition, we establish the relationship between the approximation solution and any test function in .

    Proposition 4.1. Suppose that such that on , the test function in and the approximation solution of the problem . Then, the following expression holds

    (4.3)

    where is first-order derivative order of with respect to .

    Remark 4.2. Assume that the test function , then we obtain

    (4.4)

    On the other hand, we suppose that the test function then (4.3) reads

    (4.5)

    Proof of Proposition 4.1. Assume that such that on , a test function in , then the following equation

    ()

    is well-defined. By [35, Chapter 20, Section 20.2], the problem admits a unique solution expressed in (4.3).

    Proof of Theorem 2.1 (i) We argue this proof into two steps:

    Step 1. We show that is a Cauchy sequence in a.e in . To attain this, we use the expression (4.3) to prove the Cauchy sequence. Indeed, for any there holds

    (4.6)

    From the assumption (2.12), the Eq (4.6) yields

    (4.7)

    Furthermore, by using the mean value theorem, we find that there exists a function which is continuous in such that , , where (see assumption -(i)) and are constants, therefore we obtain

    (4.8)

    whenever . By the property (2.13) of the Green function of the heat equation with nonhomogeneous Neumann boundary and the fact that , then is a constant depending on . From the Gronwall's inquality, the inequality (4.8) yields

    (4.9)

    for a.e and a constant.

    Since the sequences and are satisfying the assumption (4.1) and and are verifying the assumption (4.2), then by passing to the limit as and go to infinity, there holds

    (4.10)

    Hence the sequence is Cauchy in for almost every .

    Step 2. We show that in a.e in .

    Since the function is a solution of the approximation problem and in , in , in , then we apply the maximum principal theorem in [22,43] and then the solution of the approximation problem is nonnegative in . Likewise, we assume that , then we obtain

    (4.11)

    By the assumptions , (2.12), (2.13), (4.1) and (4.2), we infer that

    (4.12)

    By Gronwall's inequality, we deduce that

    (4.13)

    for almost every .

    By Step 1, the sequence is Cauchy in , then we infer that a.e in . Hereby we argue as in [9, Proposition 5.3], one proves that and the following convergence

    (4.14)

    for almost every holds.

    From [44, Chapter 5, Section 5.2.1, Theorem 1], the estimate (4.13) yields

    The estimate (2.10) is achieved.

    (ii) Now we show the uniqueness solutions to the problem (H).

    To attain this, we consider and two every weak solutions of the problem in sense of Definition 2.5 with initial data and respectively.

    Let be two weak solutions given by the proof (i) of Theorem 2.1. Assume that are approximating Radon measures in sense of Definition 2.4 and , in (4.11) hold. Since we have assumed that almost everywhere in , thus on . For any such that on , there holds

    (4.15)

    Let us evaluate the limit of and when . To attain this, we begin with the expression :

    Taking with such that on and , then we have

    Passing to the limit when , there holds

    (4.16)

    Now we consider the expression ,

    According to Definition 2.4, it is worth observing that

    We pass to the limit when goes to infinity, therefore

    (4.17)

    By (4.16), (4.17) and Dominated Convergence theorem, we obtain

    (4.18)

    which leads to

    Hence holds.

    Remark 4.1 (i) Since , then it is worthy observing that in (4.11) is a sequence of the approximation Radon measure satisfying the following properties

    (4.19)

    for every and is a constant.

    (ii) By (2.11)-(2.13) and the assumption (A), we deduce from the compactness theorem in [23] the approximation problem possesses a weak solution in such that the properties

    (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

    (5.1)

    Since there exists a linear continuous functional on which stands for , then we define a notion of the normal trace of the flux such that

    (5.2)

    The definition make sense because of the definition of the weak solution when we assume that the value of the initial data

    (5.3)

    holds, for any as a test function in (2.1). In particular depends only on and from (2.2), we infer the formula

    (5.4)

    for any . We denote a boundary-layer sequence of such that

    (5.5)

    For more properties concerning the boundary-layer sequence (see [37, Lemma 5.5 and Lemma 5.7]). If , then

    (5.6)

    The previous statement (5.6) explains that for any function-valued , then approaches the normal trace . Let and on , it implies that . By Definition 2.2 and the equation (5.2), the Gauss-Green formula yields

    Since and a.e in as , then Dominated Convergence Theorem ensures that

    (5.7)

    On the other hand, we consider as a test function in the problem (P) then the following expression holds

    Since and a.e in as , then Dominated Convergence Theorem yields

    (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 (resp. for any compact set ) such that (resp. and Cap (resp. ). To show that and are absolutely continuous measures with respect to the parabolic capacity, it is enough to prove that (resp. . To this purpose Cap (resp. ), there exists a sequence (resp. )) such that in (resp. in ), in (resp. in ) and in as (resp. in as ). In particular as .

    Let us consider the nonnegative function such that in and in as a test function in the problem , then there holds

    (5.9)

    By (4.3)(Probably is large enough), the following statement holds

    (5.10)

    Combining the Eq (5.9) with (5.10), we obtain

    (5.11)

    By (2.14), (see Remark 4.1-(ii)) and letting , and dropping down the nonnegative terms on the left hand-side of the previous equation. Therefore (5.11) yields

    (5.12)

    Since the following assertions are valid, then

    (5.13)

    where and

    (5.14)

    with . In view of (5.13) and (5.14), the inequality (5.12) reads as

    (5.15)

    Since (resp. ), then for any one has

    (5.16)

    Then, the limit in (5.16) as , the following holds . Therefore, for any compact set (resp. for any compact set ).

    To prove the existence and decay estimates of the solutions, we consider the following problem

    ()

    where the sequence satisfies the assumption (4.1) and the sequence fulfills the hypothesis (4.19). We set

    (5.17)

    By [8,18,21,22], the approximating problem has a solution in . Then, the definition of the weak solution of satisfies the following expression

    (5.18)

    for every in such that in and on .

    Now we establish some technical estimates which will be used in the proof of the existing solution.

    Lemma 5.2 Assume that , , , and are satisfied. Let be the solution of the approximation problem , then

    (5.19)
    (5.20)

    for almost every and is a positive constant.

    The sequence is bounded in .

    Proof of Lemma 5.2. To prove the estimate (5.19), we consider the approximation problem such that

    (5.21)

    where and , .

    Let us consider such that on and in as a test function in the above approximation problem (5.21), then we have

    (5.22)

    By the mean value theorem and the assumption (I), the Eq (5.22) yields

    (5.23)

    where .

    On the other hand, we consider the following backward parabolic equations

    (5.24)

    has an unique solution in and for any (see [18, Lemma 4.2]). Then for , there holds

    (5.25)

    By the assumptions , , (4.19) and (4.1), there exists a positive constant such that the expression below is satisfied

    (5.26)

    By letting , we obtain the estimate (5.19). Where . To prove the estimate (5.20), we consider as a test function in the approximation problem , then we have

    (5.27)

    where . It follows that there exists a positive constant such that

    For the suitable positive constant , the following estimate holds

    (5.28)

    On the other hand, we assume that and we choose as a test function in the approximation problem , then we have

    (5.29)

    It implies that

    It follows that

    (5.30)

    Combining the inequality (5.28) with (5.30), we deduce that

    (5.31)

    By the assumption (I), then , whence the estimate (5.20) holds.

    To end the proof of this Lemma, we consider that for every such that if we choose arbitrary as a test function in problem , then the following stands true

    (5.32)

    It follows that

    (5.33)

    Now we estimate each term in the right hand side of (5.33), we obtain

    (5.34)

    From Hölder's inequality and (5.31), the inequality (5.34) reads as

    (5.35)

    By the assumption and (4.19), we deduce the estimate

    (5.36)

    where is a constant.

    By the assumptions and , there exists a positive constant such that

    (5.37)

    Furthermore, one has

    In view of (5.28), the expression below holds true

    (5.38)

    where . By (5.35)-(5.38) and (5.33), we infer that the sequence is bounded in .

    Now we study the limit points of the sequences and as .

    Proposition 5.1 Suppose that the assumptions , and are satisfied. Let be the solution of the approximation problem . Then there exists a subsequence and such that

    (5.39)
    (5.40)
    (5.41)
    (5.42)

    where and .

    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 is bounded in . By [45], there exists a subsequence and such that

    Furthermore, by [9, Proposition 5.1] and (5.41) holds true and we have

    with which leads to (5.42) be satisfied. In view of the assumptions (I)- and (5.17), we get

    Therefore the following convergence holds true.

    Remark 5.1 For any subsequence and the function given in Proposition 5.1, the following assertions

    , and hold.

    Proposition 5.2 Assume that the hypotheses , , , and are satisfied. Let be the subsequence and the function mentioned in Proposition 5.1. Then there exist a subsequence and such that

    (5.43)
    (5.44)
    (5.45)

    Moreover, there hold

    (5.46)

    for almost every . Furthermore and for almost every , there holds

    (5.47)

    Proof. By the assumption (I)-, and using Hölder's inequality, we have

    From the estimate (5.20), there exists a positive constant such that

    (5.48)

    According to Lemma 5.1, the assumption (I) and (5.48), we infer that

    (5.49)

    By Fatou's Lebesgue Lemma, we obtain

    (5.50)

    Then there exists zero Lebesgue measure set such that

    (5.51)

    for every . In view of (5.51), the sequence for every . By [44, Chapter IV, Section 1.1, Proposition 5], there exists a subsequence and a.e in such that the convergence

    (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 of the Young measures associated with the sequence converges narrowly over to a Young measure which the disintegration is the Dirac mass concentrated at the point for a.e in (see [17]). By [25, Proposition A.4], there exist sequences of measure sets , and , such that

    (5.53)

    where , is a barycenter of the limiting Young measure associated with the subsequence and supp for almost every .

    By (5.19) and the compactness result, the sequence is uniformly bounded in . Therefore, there exists a Radon measure such that . Finally, the sequence is of . Hence in and the statement (5.43) is completed. By the assumption (I)-, there holds

    (5.54)

    By the assertion (5.54) and [45, Proposition 5.2] or [25], we obtain

    (5.55)

    where and

    (5.56)

    Furthermore, we also obtain the next result via (5.55)

    By combining the assertion (5.53) and the previous equality, we conclude that a.e in , when the convergence (5.44) is satisfied.

    By virtue of the convergence (5.53), the next convergence result

    (5.57)

    holds true. Since the function (see assumption (H)-(i)) and from Fatou's Lebesgue Lemma, then there exists a positive constant such that

    (5.58)

    Therefore, there exists a zero Lebesgue measure set such that

    (5.59)

    for every . In view of (5.59) and (5.57), there exists a function such that the convergence (5.45) is achieved.

    To show (5.46), we consider the functions defined by setting

    and for every . It is worthy observing that in and . According to the above results, there exists a subsequence in Lemma 5.1 and Proposition 5.1. For any nonnegative function , we choose as a test function in the approximation problem , then we obtain the following identity

    (5.60)

    where . Since the sequence is uniformly bounded in , then as and as . By Lemma 5.1 and Proposition 5.1, and by applying the Dominated Convergence Theorem, results to

    (5.61)

    Similary, we get

    (5.62)

    By the statement (4.19) and Proposition 5.1, we have

    (5.63)

    Given the properties of the sequence and passing to limit in (5.61), (5.62) and (5.63) when , then the following holds

    (5.64)

    Similarly we obtain

    (5.65)

    And

    (5.66)

    On the other hand, we have

    Since in , in and the sequence is uniformly bounded in , then we deduce that

    (5.67)

    According to the convergence statement (5.43), we have

    (5.68)

    where .

    Furthermore, from the Eqs (5.43) and (5.66) we obtain the following

    (5.69)

    It follows that

    (5.70)

    Likewise, from (5.67) one has

    It implies that

    (5.71)

    Combining the statements (5.64)-(5.66), (5.70), (5, 71) with (5.60) yields

    Since is a singular measure with respect to the Lebesgue measure for a suitable , where is zero Lebesgue measure in . Hence the assertion (5.46) is obtained.

    From [44, Chapter 5, Section 5.2.1, Theorem 1], the estimate (5.19) yields

    (5.72)

    The estimate (5.47) is completed.

    Proof of Theorem 2.2. By Proposition 5.1 and Proposition 5.2, we have a.e in . Hence the problem has a weak Radon measure-valued solution in .

    Remark 5.1 By Theorem 2.2, the result holds

    (5.73)

    for almost every . By (5.73), there exists zero Lebesgue measure set such that

    (5.74)

    for all Borel sets , with and .

    Proposition 5.3. Suppose that the assumptions and are fulfilled. Let be the subsequence and the function given in Proposition 5.1. Then the following sets

    have zero Lebesgue measure. Moreover and has zero Lebesgue measure.

    Proof of Proposition 5.3. By [9, Proposition 5.2], the set has zero Lebesgue measure. Assume that

    Then, it is worth observing that

    (5.75)

    To prove that , it is enough to show that as .

    Since the function in (see the assumption -(i)), then we have

    (5.76)

    It follows that

    (5.77)

    By the estimate (5.19), we have

    (5.78)

    Since as , then (3.62) yields as .

    Assume that , then for every . Since . Therefore, , that is holds true. The fact that , then . Consequently, is zero Lebesgue measure set.

    Proposition 6.1. Under assumptions , and . Let be a very weak Radon measure-valued solution to the problem and for every such that on , there holds

    (6.1)

    Proof of Proposition 6.1 Let us consider that for every , the smooth function , such that

    (6.2)

    Let us choose as a test function in , there holds

    (6.3)

    It is worth observing that the first term on the left hand side of the equality (6.3) gives

    (6.4)

    Let us consider a zero Lebesgue measure set in such that for any , one has

    (6.5)

    We assume that a sequence of test functions in such that

    and

    Then for every , there holds

    (6.6)

    By Dominated Convergence Theorem, we obtain

    (6.7)

    for every with

    Since , for every and for every sequence , as such that

    (6.8)

    holds true.

    Since , then we have

    (6.9)

    So that there exists a subsequence and a Radon measure such that

    (6.10)

    By the standard density arguments, one has

    (6.11)

    where , hence (6.1) is obtained.

    Proof of Theorem 2.4 Let be two very weak solutions obtained as limit of approximation of with initial data and respectively. Let , be two approximating sequence solutions to the problem . We consider a test function such that in and on in the approximation problem in the sense of the Definition 2.3, then there holds

    (6.12)

    where , , , and are two approximating functions.

    By the assumption a.e in , then for any sequences , one has on . Consequently the third term on the right hand-side of the equation (6.12) vanishes.

    For almost every , we consider the function defined as

    (6.13)

    Obviously and for every there exists a positive constant such that

    (6.14)

    This ensures that for every , the problem

    (6.15)

    has a unique solution with (see [18,21]).

    Moreover, it can be seen that

    (6.16)

    Let us consider the function such that for any and

    (6.17)

    Choosing as a test function in (6.15), then we obtain

    (6.18)

    It follows that

    (6.19)

    holds, for some constant independent on .

    From (6.16) and (6.19), there exists a constant such that

    (6.20)

    On the other hand, multiplying (6.15) by and we obtain

    which leads to

    (6.21)

    where . Therefore, we get

    (6.22)

    By standard density arguments, we can choose as a test function in (6.15). It implies that (6.12) yields

    (6.23)

    Letting to infinity in (6.23). Then it enough to observe from (6.20), there exists which is obtained by extracting the subsequence of the such that

    (6.24)
    (6.25)

    Since and the compactness theorem states in [21], we deduce that

    (6.26)
    (6.27)

    By (6.16) and (6.22), there exists such that the following statements

    (6.28)
    (6.29)

    holds true. By Theorem 2.1, the solutions of the problem are unique in . Therefore in and in . Furthermore, the sequences and satisfy the assumption (2.6). By combining the above assumptions and Dominated Convergence Theorem, the Eq (6.23) reads

    It follows that in .

    In this section, we prove the result of decay estimate solutions.

    Proof of Theorem 2.5. We consider and two solutions of the approximation problems and respectively. For any such that in and on as a test function of the approximation problem , then there holds

    (7.1)

    For every , we consider be a sequence of smooth functions such that and in . Let us choose into the Eq (7.1), then (7.1) reads

    (7.2)

    Letting in the previous equation and using the properties of the Dirac mass at , then we have the following expression

    (7.3)

    for any with and is a constant. On the other hand, by (4.5) we have

    By (2.11)-(2.13) and the properties of the Green function , we get the following result

    By the assumptions (A), (4.1) and (4.2), there exists a positive constant such that

    (7.4)

    By Gronwall's inequality, (7.4) yields

    (7.5)

    where is a constant. Combining (7.3) with (7.5), we deduce that

    (7.6)

    By [24, Chapter V, Section 5.2.1, Theorem 1], the semi-continuity of the total variation yields,

    (7.7)

    Hence (2.18) holds.

    We consider and two solutions of the approximation problems and respectively. For any such that in and on as a test function of the approximation problem . Therefore, we have the following equation

    (7.8)

    Taking into the equality (7.8), then we obtain

    Letting in the previous equation and using the properties of the Dirac mass at , then we have

    (7.9)

    By (4.1), the above inequality (7.9) yields

    (7.10)

    By [24, Chapter V, Section 5.2.1, Theorem 1], the semi-continuity of the total variation yields,

    where a constant. Hence (2.19) is achieved. Now we consider the auxiliary function such that

    (7.11)

    for every . The derivation of the expression with respect to the variable gives

    (7.12)

    Since and we multiply the Eq (7.12) by the function and then we integrate the result over (for any ), then we obtain

    (7.13)

    By replacing the expression of in (7.13), we deduce that

    (7.14)

    By assumptions , , (2.16) and (4.19), there exists a constant such that

    (7.15)

    According to [24, Chapter V, Section 5.2.1, Theorem 1], we conclude from the estimate (7.15), the following estimate

    Hence the estimate (2.21) is completed.

    To show the existence of the problem , we employ the natural approximation method. Therefore, the solution of the problem is constructed by limiting point of a family of solutions to the approximation problem. To this purpose, we consider the function such that and in (for any compact set ), then we get

    where and in with .

    Now we consider the approximation of problem

    ()

    where and a sequence of standard mollifiers. Furthermore, the sequence satisfies the assumption (4.1).

    Then for every , there exists solution of the approximation problem .

    In the next Lemma, we state the technical estimates important for the proof of the existing solutions.

    Lemma 8.1 Assume that , and are satisfied. The sequence be a weak solution of the approximation problem . Then, there holds

    (8.1)
    (8.2)

    where is a constant. Moreover, for every there holds

    (8.3)

    where is a constant.

    Proof of Lemma 8.1 We consider as a test function in the approximation problem , then we have

    (8.4)

    Assume that and . It is worth observing that . To show that in , it is enough to prove that in . Indeed, we choose , then we get

    (8.5)

    where in and in (see the assumption ). Therefore a.e in . Hence the solution of the approximation problem , a.e .

    Now we consider the regularizing sequence for every such that

    (i) in , , in ,

    (ii) as for every .

    We choose as a test function in the approximation problem and by employing the assumptions and , then we get

    (8.6)

    where . Since in (see the hypothesis ), then (8.6) reads

    (8.7)

    Again, by considering the limit when , the estimate (8.1) holds true. Now we consider another regularizing sequence for every such that if , if , if . It is obvious to see that in . We take the function and we choose as a test function in , then we obtain

    (8.8)

    Since and , therefore there exists a positive constant C such that

    (8.9)

    where . By the assumption , the statement holds. Whence the estimate (8.2) is achieved.

    Again, recalling the Hölder's inequality, we get

    where and . Therefore, there exists a positive constant such that

    (8.10)

    By the assumption , the statement holds. Hence the estimate (8.3) is achieved.

    Proof of Theorem 2.6. From the estimate (8.2) and assumption , we can extract from a subsequence such that

    (8.11)
    (8.12)

    By (8.3), the sequence and applying [44, Chapter IV, Section 1.1, Proposition 5], there exists a subsequence and such that the convergence

    (8.13)

    By repeating the same method as in the Proposition 5.2, we deduce that

    (8.14)

    where a.e in , in and a.e in .

    By [24, Chapter V, Section 5.2.1, Theorem 1], the estimate (8.1) yields

    Hence the estimate (2.23) is completed.

    Remark 8.1 The sets

    have zero Lebesgue measure. Moreover and supp.

    Proof of Theorem 2.7. We choose as a test function in the approximation problem , then we have

    By the previous proof mentioned above, we deduce that

    (8.15)

    where is a constant. By letting , then (8.15) reads

    (8.16)

    By virtue of [24, Chapter V, Section 5.2.1, Theorem 1], then the semi-continuity of the total variation yields

    (8.17)

    for almost every and . By considering to the limit as in the following inequality

    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.



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