Stability properties of Radon measure-valued solutions for a class of nonlinear parabolic equations under Neumann boundary conditions

1.   Introduction

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:

where $T > 0$, $\Omega\subset \mathbb{R}^{N}(N\geq2)$ is an open bounded domain with smooth boundary $\partial\Omega$, $\eta$ is an unit outward normal vector. The initial value data $u_{0}$ is a nonnegative bounded Radon measure on $\Omega$. The functions $\psi$ and $g$ fulfill the following assumptions

and

where $\mathbb{R}_{+}\equiv[0, +\infty)$, $\mathbb{R}_{+}^{*}\equiv(0, +\infty)$ and $\gamma\in \mathbb{R}_{+}^{*}$. By $\psi'$ and $\psi''$ we denote the first and second derivatives of the function $\psi$. The assumption (I)-$(iii)$ stem from (I)-$(i)$, hence we extend the function $\psi$ in $[0, +\infty]$ defining $\psi(+\infty) = \gamma$.

The typical example of the functions $\psi$ and $g$ are given

where $0 < m\leq 1$.

The function $h$ satisfies the following hypothesis

The function $f$ is a solution of the linear inhomogeneous heat equation under Neumann boundary conditions with measure data

where $\mu$ 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 $\Omega$, which satisfy $(P)$ in the following sense: For a suitable class of test functions $\xi$ there holds

(see Definition 2.1). Here the measure $u(\cdot, t)$ is defined for almost every $t\in (0, T)$, $u_{r}\in L^{1}(Q)$.

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],

where $\Omega_{T} = \Omega\times(0, T)$, $\sum_{T} = \sum\times(0, T)$, $(\partial\Omega\setminus\sum)_{T} = (\partial\Omega\setminus\sum)\times(0, T)$ with $\sum$ is a relative open subset of $\partial\Omega$, $\overline{\sum}$ and $\partial\Omega\setminus\sum$ are $C^{2}$ surface with boundary which meet in $C^{2}$ manifold dimension $N-2$ and $0\leq u_{0}\in L^{\infty}((\partial\Omega\setminus\sum)_{T}\cup\overline{\Omega}\times\{0\})$. 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]

where $s < t\leq T$ and $u_{0}\in W^{\tau, p}(\Omega, \mathbb{R}^{N})(\tau\in [0, \infty))$ and the definition of the operators $\mathcal{A}(t, u)u$ and $\beta(t, u)u$ are in ^[28]. Similarly, studies in ^[19,20] showed the existence and regularity of the degenerate parabolic equations with nonlinear boundary conditions and $u_{0}\in L^{2}(\Omega)$ 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 $u_{0}\in \mathcal{M}^{+}(\Omega)$ (the nonnegative bounded Radon measure on $\Omega$), secondly, the assumptions of the functions $\psi$, $g$ 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

where $\mu$ is a nonnegative Radon measure on $Q$ with $\mu(Q) < \infty$ and $\mu|_{\mathcal{R}^{N+1}\setminus Q} = 0$. 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

where $u_{0}\in \mathcal{M}^{+}(\Omega)$ and the function $\theta$ 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

where $u_{0}\in \mathcal{M}^{+}(\Omega)$ and the function $\psi$ and $g$ satisfy the hypotheses $(I)$ and $(A)$ respectively.

We consider solutions of the problem $(E)$ as maps from $\Omega$ to the cone of nonnegative bounded Radon measure on $\Omega$ which satisfies $(E)$ in the following sense: For a suitable class of test function $\varphi$, there holds

(see Definition 2.6), where $U_{r}\in L^{1}(\Omega)$ 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

where $B(U)\in L^{1}(\Omega)$, $C(U)\in L^{1}(\partial\Omega)$, $f_{2}\in L^{1}(\Omega)$, $g_{2}\in L^{1}(\partial\Omega)$ and the expression of the differential operator $L$ in [34, Section 2]. The authors proved the existence, uniqueness and regularity of the solutions $U\in W^{1, 1}(\Omega)$ 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

where $\overline{\mu}\in \mathcal{M}(\Omega)$ 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 $\psi$.

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 $L^{p}(Q)\; (1\leq p < \infty)$ 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 $L^{2}((0, T), H^{1}(\Omega))$ 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 $\Omega$.

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.

2.   Statement of the main results

To study the weak solution of the problem $(P)$, we refer to the following definition.

Definition 2.1. For any $u_{0}\in\mathcal{ M}^{+}(\Omega)$ and $\mu\in\mathcal{ M}^{+}(Q)$, a measure $u$ is called a weak solution of problem $(P)$, if $u\in \mathcal{M}^{+}(Q)$ such that

(i) $u\in L^{\infty}((0, T), \mathcal{M}^{+}(\Omega))$,

(ii) $\psi(u_{r})\in L^{2}((0, T), H^{1}(\Omega))$,

(iii) $g(u_{r})\in L^{1}(S)$,

(iv) for every $\xi\in C^{1}((0, T), C^{1}(\Omega))$, $\xi(\cdot, T) = 0$ in $\Omega$, $u$ satisfies the identity

where $u_{r}$ is the nonnegative density of the absolutely continuous part of Radon-measure with respect to the Lebesgue measure such that $u_{r}\in L^{\infty}((0, T), L^{1}(\Omega))$ and the function $f$ is the solution of the problem $(H)$.

Throughout this paper, we assume that $\Omega$ is a strong $C^{1, 1}$ open subset of $\mathbb{R}^{N}$. Also, we assume that there exists a finite open cover $(B_{j})$ such that the set $\Omega\cap B_{j}$ epigraph of a $C^{1, 1}$ function $\zeta: \mathbb{R}^{N-1}\rightarrow \mathbb{R}$ that is

where $x = (\overline{x}, x_{N})$, the local coordinates with $\overline{x} = (x_{1}, x_{2}, \ldots, x_{N-1})$. We denote $\vartheta = \{\overline{x}\, , \, x\in \Omega\cap B_{j}\}\subseteq \mathbb{R}^{N-1}$, the projection of $\Omega\cap B_{j}$ onto the $(N-1)$ first components, and $\vartheta_{\varsigma} = \{\overline{x}\, , \, x\in \text{supp}(\varsigma)\cap\Omega \}$.

If a function $\phi$ is defined on $S$, we denote $\phi_{S}$ the function defined on $(B_{j}\cap Q)\times[0, T]$ by $\xi_{S}(x, t) = \xi(\overline{x}, \zeta(\overline{x}), t)$. Notice that the restriction of $\xi_{S}$ to $[0, T)\times\vartheta$.

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 $\mathcal{F}\in [L^{2}(Q)]^{N+1}$ be such that div$\mathcal{F}$ is a bounded Radon measure on $Q$. Then there exists a linear functional $\mathcal{T}_{\eta}$ on $W^{\frac{1}{2}, 2}(S)\cap C(S)$ which represents the normal traces $\mathcal{F}\cdot\nu$ on $S$ in the sense that the following Gauss-green formula holds:

(i) For all $\xi\in C^{\infty}_{c}(\overline{Q})$,

where $\langle \mathcal{T}_{\nu}, \xi \rangle$ depends only on $\xi_{S}$.

(ii) If $(B_{j}, \varsigma, f)$ is an above subsequence localization near boundary, then for all $\xi\in C^{\infty}_{c}([0, T)\times\overline{\Omega})$ there holds

where the divergence of the fields,

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

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

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

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

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

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

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

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

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

$\mathcal{M}_{d, 2}^{+}(\Omega)$ 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

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

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

There exist two positive constants $\tau_{1}$ and $\tau_{2}$ such that

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

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

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

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.

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

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

for almost every $t\in(0, T)$.

Similarly, we consider the following problem

By Theorem 2.3, the problem $(P_{1})$ admits a solution in $L^{\infty}((0, T), \mathcal{M}^{+}(\Omega))$, such that

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

and

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

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

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

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

3.   Preliminaries

In the following section, we define the truncation function for $k > 0$ and $s\in \mathbb{R}$,

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

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

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

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

$\mathcal{M}_{s}^{+}(B)$ denotes the set of nonnegative measures singular with respect to the Lebesgue measure, namely

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

Let $\mathcal{M}_{c, 2}^{+}(B)$ be the set of nonnegative measures on $B$ which are concentrated with respect to the Newtonian capacity

$\mathcal{M}_{d, 2}^{+}(B)$ denotes the set of nonnegative measures on $B$ which are diffuse with respect to the Newtonian capacity

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

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

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

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

and

(ii) there exists a constant $C > 0$ such that

with the norm denotes

4.   Existence and Uniqueness results of the problem $(H)$

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

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

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

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

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

On the other hand, we suppose that the test function $\xi = \theta(t)\in C^{1}(0, T)$ then (4.3) reads

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

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

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

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

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

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

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

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

By Gronwall's inequality, we deduce that

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

for almost every $t\in(0, T)$ 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 $f_{1}$ 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

Let us evaluate the limit of $I_{1}$ and $I_{2}$ when $n\rightarrow \infty$. To attain this, we begin with the expression $I_{1}$:

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

Passing to the limit when $n\rightarrow \infty$, there holds

Now we consider the expression $I_{2}$,

According to Definition 2.4, it is worth observing that

We pass to the limit when $n$ goes to infinity, therefore

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

which leads to

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

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

hold true.

5.   Existence results of the problem $(P)$

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

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

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

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

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

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

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

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

On the other hand, we consider $\xi\varsigma(1-v_{\delta})$ as a test function in the problem (P) then the following expression holds

Since $0\leq v_{\delta}\leq 1$ and $v_{\delta}\rightarrow 1$ a.e in $\Omega$ as $\delta\rightarrow0^{+}$, then Dominated Convergence Theorem yields

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

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

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

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

Since the following assertions are valid, then

where $\mu^{+}(K) = \mu(K)+\mu^{-}(K)$ and

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

Since $\mu^{-}(K) = 0$ (resp. $u_{0}^{-}(K_{0}) = 0$), then for any $\epsilon > 0$ one has

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

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

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

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

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

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

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

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

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

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

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

where $T_{\lambda}(s) = \min\{\lambda, s\}$. It follows that there exists a positive constant $C$ such that

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

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

It implies that

It follows that

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

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

It follows that

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

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

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

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

Furthermore, one has

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

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

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

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

with $v = v^{*}$ which leads to (5.42) be satisfied. In view of the assumptions (I)-$(i)$ and (5.17), we get

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

Moreover, there hold

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

Proof. By the assumption (I)-$(i)$, $\psi_{n_{k}}(u_{n_{k}})\in L^{\infty}(Q)$ and using Hölder's inequality, we have

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

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

By Fatou's Lebesgue Lemma, we obtain

Then there exists zero Lebesgue measure set $\mathcal{N}_{1}\subset(0, T)$ such that

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

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

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

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

where $\psi^{*}(\cdot, t)\in L^{1}(\Omega)$ and

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

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

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

Therefore, there exists a zero Lebesgue measure set $\mathcal{N}_{2}\subseteq(0, T)$ such that

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

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

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

Similary, we get

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

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

Similarly we obtain

And

On the other hand, we have

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

According to the convergence statement (5.43), we have

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

It follows that

Likewise, from (5.67) one has

It implies that

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

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

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

for almost every $t\in(0, T)$. By (5.73), there exists zero Lebesgue measure set $\mathcal{N}_{3}\subset(0, T)$ such that

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

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

Then, it is worth observing that

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

It follows that

By the estimate (5.19), we have

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.

6.   Uniqueness results of the problem $(P)$

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

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

Let us choose $\rho_{j}(x)\eta_{\tau}(t)$ as a test function in $(P)$, there holds

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

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

We assume that a sequence $\{\rho_{j}(x)\}$ of test functions in $\Omega$ such that

and

Then for every $t\in(0, T)\setminus D_{j}$, there holds

By Dominated Convergence Theorem, we obtain

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

holds true.

Since $u\in L^{\infty}((0, T), \mathcal{M}^{+}(\Omega))$, then we have

So that there exists a subsequence $\{t_{j_{m}}\}\subseteq \{t_{j}\}$ and a Radon measure $\mu_{0}\in \mathcal{M}^{+}(\Omega)$ such that

By the standard density arguments, one has

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

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

Obviously $a_{n}\in L^{\infty}(Q)$ and for every $n\in \mathbb{N}$ there exists a positive constant $C_{n}$ such that

This ensures that for every $z\in C^{2}_{c}(Q)$, the problem

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

Let us consider the function $\beta$ such that for any $t_{1}+1 < t_{2}$ and $t_{1}\;, \; t_{2}\in (0, T)$

Choosing $\beta\Delta\xi_{n}$ as a test function in (6.15), then we obtain

It follows that

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

On the other hand, multiplying (6.15) by $\Delta\xi_{n}$ and we obtain

which leads to

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

By standard density arguments, we can choose $\xi = \xi_{n}$ as a test function in (6.15). It implies that (6.12) yields

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

Since $\xi_{nt}\in L^{2}(Q)$ and the compactness theorem states in ^[21], we deduce that

By (6.16) and (6.22), there exists $\xi(\cdot, 0)\in L^{\infty}(\Omega)\cap H^{1}(\Omega)$ such that the following statements

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

It follows that $u_{1} = u_{2}$ in $\mathcal{M}^{+}(Q)$.

7.   Decay estimates solutions of the problem $(P)$

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

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

Letting $\epsilon\rightarrow 0^{+}$ in the previous equation and using the properties of the Dirac mass at $t$, then we have the following expression

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

By (2.11)-(2.13) and the properties of the Green function $G_{N}$, we get the following result

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

By Gronwall's inequality, (7.4) yields

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

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

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

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

Letting $\epsilon\rightarrow 0^{+}$ in the previous equation and using the properties of the Dirac mass at $t$, then we have

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

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

where $C = C(T, \alpha) > 0$ a constant. Hence (2.19) is achieved. Now we consider the auxiliary function $\mathcal{W}_{n}$ such that

for every $\alpha > 1$. The derivation of the expression $\mathcal{W}_{n}$ with respect to the variable $t$ gives

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

By replacing the expression of $\mathcal{W}_{n}$ in (7.13), we deduce that

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

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.

8.   Asymptotic behavior solutions of the problem $(P)$

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

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 $(E)$

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

where $C > 0$ is a constant. Moreover, for every $1\leq p < \frac{N}{N-1}$ there holds

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

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

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

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

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

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

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

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

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

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

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

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

Hence the estimate (2.23) is completed.

Remark 8.1 The sets

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

By the previous proof mentioned above, we deduce that

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

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

for almost every $t\in(0, T)$ and $\alpha > 1$. By considering to the limit as $t\rightarrow +\infty$ in the following inequality

Hence the statement (2.24) follows.

9.   Conclusions

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).

Acknowledgment

This work was partially supported by National Natural Sciences Foundation of China, grant No: 11571057

Conflict of interest

The authors declare no conflict of interest.