Abstract
In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows:
where \(T>0\), \(\Omega \subset \mathbb{R}\) is a bounded interval, \(u_{0}\) is nonnegative bounded Radon measure on Ω, and \(\alpha , \beta \geq 0\), under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.
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1 Introduction
In this paper we address the existence, decay estimate, and the asymptotic behavior of solutions for the following problem:
where \(T>0\), \(\Omega \subset \mathbb{R}\) is a bounded interval, \(u_{0}\) is nonnegative bounded Radon measure on Ω under suitable assumptions on ψ and f expressed as follows:
for any \((\alpha ,\beta )\in \mathbb{R}_{+}\times \mathbb{R}_{+}\), where the function φ satisfies the following assumptions:
where \(\mathbb{R}_{+}\equiv [0,+\infty )\), \(\mathbb{R}_{+}^{*}\equiv (0,+\infty )\), and \(\gamma \in (0,+\infty )\). A typical example of the function φ is given by
where \(0< m\leq 1\), \(s\geq 0\).
The function f verifies the following assumption:
The example of the function f is
The assumption of the function ψ in (1.1) and hypothesis (H) are summarized as follows:
where \(\psi '\) and \(\psi ''\) are the first derivative and the second derivative of the function ψ respectively. Similarly, hypothesis (G) is inferred from assumption (H) and the expression of the function ψ given by (1.1). Since \(\psi (s)\rightarrow +\infty \) as \(s\rightarrow +\infty \), problem (P) is not degenerate at infinity. However, problem (P) is degenerate at zero.
Remark 1.1
By assumption (R) and the mean-value theorem, it is worthy observing that
Then, for every \(\epsilon >0\), there exists \(M=M(\epsilon )>0\) such that
By the example of the function f in (1.2)–(1.3), it implies \(f'(+\infty )=0\). We notice that the function f can also satisfy assumption (R) with \(f'(+\infty )\neq 0\).
Throughout this paper, we consider the solutions of problem (P) as maps from \((0,T)\) to the cone of nonnegative bounded Radon measure on Ω, which verify (P) in the following sense: for a suitable class of test function ξ, there holds
where \(\overline{\xi }_{\nu }=\xi _{t}+\alpha \xi _{xx}\) (see Definition 2.1). Here the measure \(u(\cdot ,t)\) is defined for a.e. \(t\in (0,T)\), \(u_{r}\in L^{1}(Q)\).
The one-dimensional degenerate parabolic equations with initial data have been intensively investigated in several decades. Similarly, the nonlinear parabolic equation (P) has been studied by many authors such as (see [2–4, 6, 32]). Within the type of problem (P), we consider the problem studied in [2]:
where \(L>0\) and the functions ϕ̃ and h verify the suitable assumptions (see [2] for more details). In [2], the authors studied the existence, uniqueness, and regularity of the solutions to problem (A.1) when \(u_{0}\in L^{\infty }(-L,L)\). Meanwhile, the main purpose of the study of problem (A.1) is the convergence of the solutions when \(t\rightarrow +\infty \). The difference between problem (A.1), the references [3, 4, 6, 32], and problem (P) is the assumptions which satisfy the functions ψ, f and the initial data \(u_{0}\).
In the literature of one-dimensional nonlinear parabolic equations with initial data, there are many studies of the kind of problem (P) without the source term (\(f(u)=0\)) (for instance, see [5, 11, 28, 29, 31, 33–40]). In [28], the authors studied the following problem:
where \(T>0\), \(\Omega \subset \mathbb{R}\) is a bounded interval, \(u_{0}\) is nonnegative bounded Radon measure on Ω, and the function ϕ is nonmonotone and satisfies the hypothesis including (H) (see [28] for more details).
In [28], the authors dealt with the existence of the weak Radon measure-valued solutions associated with the Young measure. The difference between problem (A.2) [5, 11, 29, 31, 33–40] and problem (P) remains the hypothesis on the function f given by (R).
In [14], the authors addressed the existence, uniqueness, and the qualitative properties of the Radon measure-valued solutions associated with the Young measure to the first order scalar conservation laws with Radon measure as initial data. The problem studied in [14] is
where \(u_{0}\) is nonnegative bounded Radon measure on Ω and the function g verifies hypothesis (G).
Before we study the general problem of (P), we need to point out the particular cases of such a problem and their results. For instance \(\alpha =0\), problem (P) is nonlinear degenerate parabolic equations, and this kind of nonlinear degenerate parabolic equations is similar to
where \(T>0\), \(\Omega \subset \mathbb{R}^{N} (N\geq 2)\) is an open bounded domain with smooth boundary ∂Ω and \(u_{0}\) is a finite Radon measure on Ω. The operator \(A(x,t,s)\) is weakly coercive and diffuse and \(F(x,t,u)\) verifies the suitable hypothesis (see [1] for more details). In [1], the authors proved the existence and qualitative properties of the Radon measure-valued solutions associated with the Young measure. Another difference between problem (A.4) and (P) is the assumption which fulfills the function \(F(x,t,u)\). Indeed, the hypothesis of the function \(F(x,t,u)\) is different from assumption (R) (\(f(u)\) verifies hypothesis (R) of problem (P)). On the other hand, the problems studied in the papers [44–46] are closely formulated as in (A.4), where the expression for the source term \(F(x,t,u)\) is more regular and the diffusion-term \(A(x,t,s)\) takes part on the modeling of real phenomena from mathematical biology and physics. Furthermore, the authors in [44–46] dealt with the properties of weak and classical solutions.
Assuming that \(\beta =0\), problem (P) is reduced to the semilinear heat equation with Radon measure as initial data described as follows:
where \(\alpha \equiv 1\), \(T>0\), \(\Omega \subset \mathbb{R}\) is a bounded interval, \(u_{0}\) is nonnegative bounded Radon measure on Ω. By [19, 20], problem (A.5) admits unique weak solutions which are not Radon measure-valued associated with the Young measure. However, in [17] the authors showed the existence, qualitative properties, and decay estimate of the Radon measure-valued solutions to the Cauchy problem of (A.5). Throughout this paper, we consider the case \(\alpha >0\) and \(\beta >0\), and we notice that the result of this paper is not true for \(\alpha =0\).
The goal of this paper is threefold. Firstly, we study the existence of the Radon measure-valued solutions associated with the Young measure introduced in [1] and the other technical tools stated in [14, 28].
Secondly, we establish the decay estimate of the Radon measure-valued solutions to problem (P). We note that the proof of the existence of the Radon measure-valued solutions and the decay estimate of these weak solutions focus on the natural approximation method.
Thirdly, we analyze the asymptotic behavior of the Radon measure-valued solutions. To this purpose, we construct the pseudo-stationary solutions which are Radon measure-valued solutions to the nonlinear elliptic equations. Then the result of the asymptotic behavior of solutions follows from the use of the natural approximation method.
The novelty of this paper is twofold. Firstly, we study the decay estimate of the Radon measure-valued solutions of a class of nonlinear parabolic equations. Finally, we study the asymptotic behavior of these Radon measure-valued solutions.
The plan of this paper is organized as follows. In the next section, we recall some preliminaries about Radon measures and Young measures. Then, in Sect. 3, we state the main results, while in Sects. 4–7 we prove the main results.
2 Preliminaries
2.1 Radon measures
Let \(\mathcal{M}(\Omega )\) be the space of bounded Radon measures on Ω, and \(\mathcal{M}^{+}(\Omega )\subset \mathcal{M}(\Omega )\) be the cone of nonnegative bounded Radon measure on \(\Omega \subseteq \mathbb{R}^{N} (N\geq 1)\). For any \(\mu \in \mathcal{M}(\Omega )\), we set
where \(| \mu | \) stands for the total variation of μ.
The duality map \(\langle \cdot ,\cdot \rangle _{\Omega }\) between the space \(\mathcal{M}(\Omega )\) and \(C_{c}(\Omega )\) is defined by
For any \(\mu \in \mathcal{M}(\Omega )\) and any Borel set \(B\subseteq \Omega \), the restriction \(\mu \llcorner B\) of μ to B is defined by setting
It is worth observing that \((\mu \llcorner B )(\emptyset )=0\).
\(\mathcal{M}_{s}^{+}(\Omega )\) denotes the set of nonnegative measures singular with respect to the Lebesgue measure, namely
\(| \cdot| \) denotes the Lebesgue measure on \(\mathbb{R}^{N}(N\geq 1)\). Similarly, \(\mathcal{M}_{ac}^{+}(\Omega )\) denotes the set of nonnegative measures absolutely continuous with respect to the Lebesgue measure, namely
Recall that \(\mathcal{M}_{s}^{+}(\Omega )\cap \mathcal{M}_{ac}^{+}(\Omega )=\{0\}\). Moreover, by the Lebesgue decomposition and Radon–Nikodym theorem (see [9]), for any \(\mu \in \mathcal{M}^{+}(\Omega )\):
(i) There exists a unique couple \(\mu _{ac}\in \mathcal{M}_{ac}^{+}(\Omega )\), \(\mu _{s}\in \mathcal{M}_{s}^{+}(\Omega )\) such that
(ii) There exists a unique nonnegative function \(u_{r}\in L^{1}(\Omega )\) called the density of measure \(\mu _{ac}\) such that
Let \(Q=\Omega \times (0,T)\), T is a positive constant.
\(L^{\infty } ((0,T),\mathcal{M}^{+}(\Omega ) )\) denotes the set of nonnegative Radon measures \(u\in \mathcal{M}^{+}(Q)\) which satisfy the following property: For almost every \(t\in (0,T)\), there exists a measure \(u(\cdot ,t)\in \mathcal{M}^{+}(\Omega )\) such that
(a) For every \(\xi \in C(\overline{Q})\), the map \(t\mapsto \langle u(\cdot ,t),\xi (\cdot ,t)\rangle _{\Omega }\) is Lebesgue measurable and there holds
(b) For every Borel set \(B\subseteq Q\), the map \(t\mapsto u(\cdot ,t)(B^{t})\) is Lebesgue measurable and there holds
where \(B^{t}= \{ x\in \Omega / (x,t)\in B \} \).
(c) There exists a constant \(C>0\) such that
Set
If \(u\in L^{\infty }((0,T),\mathcal{M}^{+}(\Omega ))\), it is easily seen that \(u_{ac}, u_{s}\in L^{\infty }((0,T),\mathcal{M}^{+}(\Omega ))\) as well and that \(u_{r}\in L^{\infty }((0,T),L^{1}(\Omega ))\).
Moreover, inequality (2.4) implies that, for every \(\xi \in C(\overline{Q})\),
2.2 Young measures
We denote by \(C_{c}(\mathbb{R})\) the space of continuous real functionals with compact support in \(\mathbb{R}\) and by \(\mathcal{M}(\mathbb{R})\) the Banach space Radon measure on \(\mathbb{R}\) endowed with the norm
By \(\mathcal{M}(\mathbb{R})\) we denote the cone of positive finite Radon measure, and by \(\mathcal{P}(\mathbb{R})\) the convex set of probability measure on \(\mathbb{R}\):
Clearly, \(\mathcal{P}(\mathbb{R})\subset \mathcal{M}^{+}(\mathbb{R})\subset \mathcal{M}(\mathbb{R})\).
By a bounded Caratheodory integrand on \(A\times \mathbb{R}\) we mean that any function \(\varphi :A\times \mathbb{R}^{N}\rightarrow \mathbb{R}\) is bounded and measurable, with \(\varphi (x,\cdot )\) continuous for almost everywhere \(x\in A\). The duality map \(\langle \cdot ,\cdot \rangle \) between the spaces \(\mathcal{M}(\mathbb{R})\) and \(C_{c}(\mathbb{R})\) is expressed as
which can be extended to functions \(\rho \in C_{c}(\mathbb{R})\). Let \(A\subset \mathbb{R}^{N}(N\geq 1)\) be a bounded open set. We use the above equality to define the quantity \(\langle \mu ,\rho \rangle _{\mathbb{R}}\) for any \(\mu \in \mathcal{M}(\mathbb{R})\) and every μ-integrable function ρ. Similar notation will be used for the space \(\mathcal{M}(A\times \mathbb{R})\) of finite Radon-measures on \(A\times \mathbb{R}\). By \(\mathcal{Y}(A,\mathbb{R})\) we denote the set of Young measures on \(A\times \mathbb{R}\) which are defined as follows (e.g. [15]).
Definition 2.1
A Young measure on \(A\times \mathbb{R}\) is a positive Radon-measure τ on \(A\times \mathbb{R}\) such that
If \(f\in L^{1}(A)\), the Young measure associated with f is the measure \(\tau \in \mathcal{Y}(A,\mathbb{R})\) such that
For any bounded Caratheodory integrand, there holds
Let us recall the following result (e.g. [9, 10, 15]).
Proposition 2.1
Let \(\tau \in \mathcal{Y}(A,\mathbb{R})\). Then, for almost everywhere \(x\in A\), there exists a probability measure \(\tau _{x}\in \mathcal{P}(\mathbb{R})\) for any bounded Caratheodory integrand φ on \(A\times \mathbb{R}\):
(i) The map
is Lebesgue measurable;
(ii) There holds
More generally, Proposition 2.1 holds true for very \(\varphi :A\times \mathbb{R}\rightarrow \mathbb{R}\) measurable and nonnegative or τ-integrable, we shall identify and \(\tau \in \mathcal{Y}(A,\mathbb{R})\) with the associated family \(\{ \tau _{x}| x\in A \} \) which is called disintegration of τ.
Remark 2.1
If τ is the Young measure associated with a function \(f\in L^{1}(A)\), equalities (2.8)–(2.9) imply that
for almost every \(x\in A\) and for any bounded Caratheodory integrand φ on \(A\times \mathbb{R}\). Therefore,
where \(\delta _{p}\) denotes the Dirac mass concentrated in \(p\in \mathbb{R}\).
The notion of narrow convergence of Young measures is as follows.
Definition 2.2
Let \(\tau ^{n}\), \(\tau \in \mathcal{Y}(A,\mathbb{R})\). We say that \(\tau ^{n}\rightarrow \tau \) narrowly in \(A\times \mathbb{R}\) if for any bounded Caratheodory integrand \(\varphi :A\times \mathbb{R}\rightarrow \mathbb{R}\) there holds
Remark 2.2
If the Young measures \(\{ \tau ^{n} \} \) are associated with a sequence of functions \(\{ f_{n} \} \subseteq L^{1}(A)\), then \(\{ \tau ^{n} \} \) converge narrowly to τ if and only if
for any bounded Caratheodory integrand.
This convergence still holds when φ is a Caratheodory integrand with linear growth with respect to ξ (i.e. \(| \varphi (x,\xi )| \leq \alpha (x)+| \xi | \), where \(\alpha \in L^{1}(A)\) as soon as the sequence \(\{ f_{n} \} \) is uniformly integrable.
If \(\tau ^{n}\) and τ are the Young measures associated with the measurable functions \(f_{n}\) and f respectively, then \(\tau ^{n}\rightarrow \tau \) narrowly if and only if \(f_{n}\rightarrow f\) in measure. In other words, \(f_{n}\rightarrow f\) in measure if and only if the Young measure associated with \(f_{n}\) is \(\delta _{f(x)}\) (see [15, 16]).
Definition 2.3
A subset \(\mathcal{U}\subseteq L^{1}(A,\mathbb{R})\) is said to be uniformly integrable if
(i) There exists \(M>0\) such that
(ii) For \(\epsilon >0\), there exists \(\beta >0\) such that, for any \(f\in \mathcal{U}\),
Notice that for any function \(f\in L^{1}(A)\) is equi-integrable if the assumption of Definition 2.3-(ii) holds true.
The following proposition is a consequence of the more general Prokhorov’s theorem (e.g. see [15]).
Proposition 2.2
Let \(\{f_{n}\}\) be bounded in \(L^{1}(A)\) and \(\{\tau ^{n}\}\) be the sequence of associated Young measures. Then
(i) There exist a sequence \(\{f_{n_{k}}\}\subseteq \{f_{n}\}\) and a Young measure τ on \(A\times \mathbb{R}\) such that \(\tau ^{n_{k}}\rightarrow \tau \) narrowly in \(A\times \mathbb{R}\);
(ii) For any \(h\in C(\mathbb{R})\) such that the sequence \(\{h(f_{n_{k}})\}\) is bounded in \(L^{1}(A)\) and is equi-integrable, there hold
where
\(\tau _{x}\) is disintegration of the Young measure τ.
(iii) For any function \(\varphi :A\times \mathbb{R}\rightarrow \mathbb{R}\) measurable with \(\varphi (x,\cdot )\) continuous for a.e., \(x\in A\) such that the sequence \(\{\varphi (x,f_{n_{k}} )\}\) is bounded in \(L^{1}(A)\) and is equi-integrable, there holds
When equi-integrability of the sequence \(\{f_{n_{k}}\}\) fails, Proposition 2.2-(ii) cannot be directly used with \(h(f)=f\). However, we can associate with \(\{f_{n_{k}}\}\) an equi-integrable subsequence by removing sets of small measure, this is the content of the next coming proposition (e.g. see [10, 15]).
Proposition 2.3
(Biting Young measure)
Let \(\{f_{n}\}\) be bounded in \(L^{1}(A)\). Let \(\tau \in \mathcal{Y}(A,\mathbb{R})\) and \(\{f_{n_{k}}\}\) be respectively the limiting Young measure and a subsequence given in Proposition 2.3in correspondence with \(\{f_{n}\}\). Then there exist a subsequence \(\{f_{n_{j}}\}\equiv \{f_{n_{k_{j}}}\}\subseteq \{f_{n_{k}}\}\) and a sequence of measurable sets \(\{A_{j}\}\), \(A_{j}\subset A\), \(A_{j+1}\subset A_{j}\) for any \(j\in \mathbb{N}\), \(| A_{j}| \rightarrow 0\) as \(j\rightarrow \infty \) such that the sequence \(\{f_{n_{j}}\chi _{A\backslash A_{j}}\}\) is equi-integrable. Moreover, the barycenter of the Young measure disintegration \(\tau _{x}\),
3 Statement of main results
Throughout this paper, we consider the backward parabolic equation
which has a unique solution ξ in \(C^{1,2}(Q)\cap C^{1}(\overline{Q})\) for any \(\xi _{\nu }\in C(Q)\) (see [12, 13, 30]), where
Definition 3.1
For any \(u_{0}\in \mathcal{ M}^{+}(\Omega )\), a function u is called a weak Radon measure solution of problem (P) if the couple \((u,\tau )\) is such that
(i) \(u\in L^{\infty }((0,T), \mathcal{ M}^{+}(\Omega ))\), \(\tau \in \mathcal{Y}(Q,\mathbb{R})\)
(ii) \(\psi (u_{r})\in L^{1}((0,T), W^{1,1}_{0}(\Omega ))\);
(iii) For almost everywhere \((x,t)\in Q\), there hold
where we denote by \(\delta _{u_{r}(x,t)}\) the Dirac mass concentrated at \(u_{r}(x,t)\) and \(\tau _{(x,t)}\in \mathcal{P}(\mathbb{R})\) is the disintegration of τ.
(iv) For every \(\xi \in C^{1}([0,T], C^{1}_{0}(\Omega ))\), \(\xi (\cdot ,T)=0\) in Ω, u satisfies the identity
whenever \(\overline{\xi }_{\nu }=\xi _{t}+\alpha \xi _{xx}\) and
for a.e. \((x,t)\in Q\).
Remark 3.1
Since the class of test functions \(\xi \in C^{1}([0,T], C^{1}_{0}(\Omega ))\) is a solution to the backward parabolic equations \((\nu \cdot \alpha )\) such that \(\xi _{\nu }\in C(Q)\), then Eq. (3.3) is reduced as follows:
where \(\xi _{\nu }=\xi _{t}+\alpha \xi _{xx}+f'(+\infty )\xi \) in Q.
The existence solution to problem (P) is given by the following result.
Theorem 3.1
Suppose that (1.1), (1.4), (H), (R), (G), and \(u_{0}\in \mathcal{M}^{+}(\Omega )\) are satisfied. Then problem (P) has a solution \((u,\tau )\) which is obtained as a limit point of the sequence \(\{u_{n}\}\) of solutions to problem \((P_{n})\). Moreover, \((u,\tau )\) is a solution of problem (P) in the sense of Young measures.
Let us consider the following problem:
In view of [12, 13], problem (Z) has a unique solution function v in \(L^{\infty }(Q)\cap L^{2}((0,T), H^{1}_{0}(\Omega ))\cap C(Q)\). Moreover, from Theorem 3.1, problem (Z) possesses a weak solution \(v(\cdot ,t)\in \mathcal{M}^{+}(\Omega )\) for a.e. \(t\in (0,T)\). The estimate decay is given by the following result.
Theorem 3.2
Suppose that (R), (G), and \(u_{0}\in \mathcal{M}^{+}(\Omega )\) are verified. The functions u and v are weak solutions to problems (P) and (Z) respectively in the sense of Theorem 3.1. Then there holds
for any \(\alpha >0\) and C is a positive constant. Moreover, if we extend \(t\in (0,T)\) into \((0,+\infty )\), then we obtain that the following statement
holds true.
Regarding the study of the asymptotic behavior of the Radon measure-valued solutions to the nonlinear parabolic equation (P), we construct the pseudo-stationary solutions to problem (P). To this purpose, we consider the pseudo-stationary problem as follows:
where \(u_{0}\in \mathcal{M}^{+}(\Omega )\) and the function ψ verifies hypothesis (G). Notice that the nonlinear elliptic equation (S) admits a nonnegative Radon measure-valued solution i.e. \(w\in \mathcal{M}^{+}(\Omega )\).
The main goal of the asymptotic behavior of the Radon measure-valued solutions to problem (P) is given in the following theorem.
Theorem 3.3
Assume that hypotheses (1.1), (R), (G), and \(u_{0}\in \mathcal{M}^{+}(\Omega )\) are fulfilled. w is a pseudo-stationary Radon measure-valued solution obtained from problem (S) and u is a Radon measure-valued solution in the sense of Theorem 3.1such that
Then
Remark 3.2
Before we prove assertion (3.8), we shall ensure that statement (3.7) holds true. Indeed, the function u is a Radon measure-valued solution to problem (P), then \(u(\cdot ,t)\in \mathcal{M}^{+}(\Omega )\) for a.e. \(t\in (0,T)\). By the extension of the solution to global solutions on \(\Omega \times (0,+\infty )\), we infer that \(\| u(\cdot ,t)\| _{\mathcal{M}^{+}(\Omega )}\leq C\) for a.e. \(t\in (0,+\infty )\) so that (3.8) is obtained.
4 Approximating problems
To prove the existence, decay estimate, and the asymptotic behavior of the solutions, we consider the approximating problem \(P_{n}\) as follows:
The approximating function \(\psi _{n}\) is such that
Since \(u_{0}\in \mathcal{M}^{+}(\Omega )\), then the approximation of the Radon measure \(u_{0}\) is given by [7, Lemma 4.1], such that \(\{u_{0n}\}\subseteq C_{0}^{\infty }(\Omega )\) satisfies
Notice that the use of \(L^{1}\)- norm of the initial data is a consequence of the above [7, Lemma 4.1].
The existence of a weak solution \(u_{n}\) in \(C((0,T),L^{1}(\Omega ))\cap L^{2}((0,T),H^{1}_{0}(\Omega ))\cap L^{ \infty }(Q)\) of problem \((P_{n})\) is ensured by [4, Theorem 5] and [21, Chapter V, Theorem 2.1 and 6.7]. Moreover, \(u_{nt}\in L^{2}((0,T),H^{-1}(\Omega ))\) given by [22, Proposition 6.1] and \(u_{n}\in L^{\infty }(Q)\) is proved in [23, Theorem 3.1]. Then a definition of the weak solution \(\{u_{n}\}\subseteq C^{\infty }(\overline{Q})\) of \((P_{n})\) satisfies the following statement:
for every ξ in \(C^{1}(\overline{Q})\) such that \(\xi (\cdot ,T)=0\) and \(\xi =0\) on \(\partial \Omega \times (0,T)\).
Now we establish some technical estimates which will be used in the proof of the existence solution.
Proposition 4.1
Suppose that \(f\geq 0\) in \(\mathbb{R}_{+}\) and (G) holds. Moreover, \(u_{0n}\geq 0\) in Ω. Then there holds
Proof
Since \(\psi (u_{n})=\alpha u_{n}+\beta \varphi (u_{n})\). Let us consider the nonlinear parabolic boundary value problem
According to [21, Chapter V, Theorem 2.1 and Theorem 6.7], problem \(P_{n}\) possesses a unique solution \(u_{n}\). Since \(u_{0n}\geq 0\) in Ω, and \(f(u_{n})\geq 0\) in Q, it follows that
By [25, Chap. 10, Theorem 10.1], the comparison maximum principle theorem, we obtain \(u_{n}(x,t)\geq 0\) in Q̅, whence estimate (4.4) holds. □
Lemma 4.1
Assume that \((R)\) and (G) are satisfied and \(u_{0}\in \mathcal{M}^{+}(\Omega )\). For every \(t\in [0,T]\), there holds
where \(L:=\| f'(u_{n})\| _{L^{\infty }(\mathbb{R}_{+})}\) is a positive constant.
Proof
Let us consider the boundary value problem
By Proposition 4.1, \(\psi (u_{n})\geq 0\) in Q and \(\psi (u_{n})=0\) on \(\partial \Omega \times (0,T)\). In view of [21, Chapter V], the maximum principle theorem, then for arbitrary point \((x_{0},t_{0})\in \partial \Omega \times (0,T)\), we obtain
the normal outer derivative of \(\psi (u_{n})(x_{0},t_{0})\) at \(\partial \Omega \times (0,T)\). Applying Green’s formula for every \((x,t)\in Q\), there holds
where \(\mathcal{H}\) denotes the Hausdorff \((N-1)\)-dimensional measure. Using the Eq. (4.6), assumption (4.2), and the mean-value theorem, we deduce that
By Gronwall’s inequality, estimate (4.5) is achieved. □
Proposition 4.2
Assume that hypotheses (R) and (G) are satisfied. Let \(u_{n}\) be the solution of \((P_{n})\). Then there hold
Proof
By the definition of ψ in (1.1), \(\psi (u_{n})=\alpha u_{n}+\beta \varphi (u_{n})\). By (4.5), \(u_{n}\) is bounded in \(L^{1}(Q)\), and assumption (G), it is obvious that there exists a constant \(C>0\) such that (4.8) is achieved.
Since \(\psi (u_{n})\geq 0\) in Q and \(\psi (u_{n})=0\) on \(\partial \Omega \times (0,T)\). The fact that \(u_{n}=\psi (\psi ^{-1}(u_{n}))\in C^{1}([0,T],H^{1}_{0}(\Omega ))\). Let us consider the function sign defined in the following manner:
Assume that, for every \(K>0\), \(T_{K}\in C^{1}(\mathbb{R}_{+})\cap L^{\infty }(\mathbb{R}_{+})\) such that \(1\leq T'_{K}(s)\leq 2\) in \(\mathbb{R}_{+}\), \(T_{K}(0)=0\) and \(T_{K}(s)\rightarrow \frac{3}{2}s\chi _{(1,1+\epsilon )}(s)\) as \(K\rightarrow +\infty \) for every \(0<\epsilon <1\), where \(\chi _{(1,1+\epsilon )}\) is a characteristic function on \((1,1+\epsilon )\).
For instance, let us consider \(\{g_{K}\}\subseteq C^{1}(\mathbb{R})\) such that \(g_{K}(s)\rightarrow \)\(\operatorname{sign} (s)\) as \(K\rightarrow +\infty \) for every \(s\neq 0\) to be any sequence satisfying the following conditions: \(g_{K}(0)=0\), \(| g_{K}(s)| \leq 1\), \(g'_{K}(s)\geq 0\), \(| sg_{K}(s)| \leq 1\) for every \(s\in \mathbb{R}\), and \(g'_{K}(s)=0\) if \(| s| \geq \frac{1}{K}\). By recalling the sequence \(\{g_{K}\}\) constructed in [1], we can construct the sequence of the function \(\{T_{K}\}\subseteq C^{1}(\mathbb{R}_{+})\cap L^{\infty }(\mathbb{R}_{+})\) such that \(T_{K}(s)=(1+\frac{1}{2}g_{K}(s))s\chi _{(1,1+\epsilon )}(s)\).
Assume that \(T_{K}(\psi (u_{n}))\) is a test function to the approximation problem \((P_{n})\). Then we get
Since \(1\leq T'_{K}(\psi (u_{n}))\leq 2\) and \(T_{K}(\psi (u_{n}))\), \(T_{K}(\psi (s))\in L^{\infty }(\mathbb{R}_{+})\) for every K, then (4.12) yields
Then there exists a positive constant \(C=C (\| u_{0}\| _{\mathcal{M}^{+}(\Omega )}, \| f(u_{n})\| _{L^{\infty }(\mathbb{R}_{+})}, | Q | )>0\) such that
Assume that \(\xi \psi '(u_{n})\) is a test function into the first equation of \((P_{n})\). Then we have
for any \(\xi \in C^{1}_{c}(Q)\). Let us estimate each term of the right-hand side of Eq. (4.13). To this purpose, we consider its first term
By hypothesis (G), \(\psi '(u_{n})\), \(\psi ''(u_{n})\in L^{\infty }(\mathbb{R}_{+})\), then we obtain
By (4.9) and applying Holder’s inequality, there exist two positive constants \(C_{0}\) and \(C_{1}\) such that
On the other hand, we consider the second term on the right-hand side of Eq. (4.13). From assumptions (G) and (R), we deduce that
where \(C_{2}\) is a positive constant. Hence the boundedness of the sequence \(\{ [\psi (u_{n})]_{t} \} \) in \(L^{2}((0,T),H^{-1}(\Omega ))+L^{1}(Q)\) follows.
To end this proof, it remains to establish estimate (4.11), let us consider the function h defined by
for any \(t_{1} , t_{2}\in (0,T)\) such that \(t_{1}+1< t_{2}\), and we observe that \(0\leq h(s)< C\), \(s\in (0,t_{2})\). By assumption (G), it is easy to observe that for every \(n\in \mathbb{N}\) \([\psi (u_{n})]_{s}=0\) on \(\partial \Omega \times (0,T)\) for the homogeneous Dirichlet condition. Multiplying the approximation problem \((P_{n})\) by the test function \(h(s)[\psi (u_{n})]_{s}\) and integrating over \(\Omega \times (0,t_{2})\), we obtain
It implies that
In view of assumption (R), \(f(u_{n})\in L^{\infty }(\mathbb{R}_{+})\), and the fact that (4.10) is satisfied, the last term on the right-hand side of the previous estimate (4.18) is bounded. Since (4.9) holds, there is a positive constant C such that (4.11) is achieved. □
5 Existence result
Now we study the limit points of the sequences \(\{u_{n}\}\) and \(\psi (u_{n})\) as \(n\rightarrow \infty \).
Proposition 5.1
Suppose that (1.1), \((R)\), and (G) are satisfied. Let \(u_{n}\) be the solution of the approximating problem \((P_{n})\). Then there exist a subsequence \(\{u_{n_{k}}\}\subseteq \{ u_{n}\}\), \(v\in L^{2}((0,T),H^{1}_{0}(\Omega ))\cap L^{\infty }((0,T),H^{1}_{0}( \Omega ))\cap L^{\infty }(Q)\) and \(v_{t}\in L^{2}(Q)\) such that
Proof
By using Holder’s inequality
From estimate (4.9), there exists a positive constant \(C>0\) such that
According to Proposition 4.2, assumption (G), and (4.8), we infer that
By [43, Chap. 3, Sect. 3.1, Theorem 3.23, and Theorem 3.9], convergence (5.1) holds. However, convergence (5.2) is the consequence of estimate (4.9). By Proposition 4.2, the sequence \(\{ [\psi (u_{n})]_{t} \} \) is bounded in \(L^{2}((0,T),H^{-1}(\Omega ))+L^{1}(Q)\), then there exist a subsequence denoted again \(\{u_{n_{k}}\}\subseteq \{ u_{n}\}\) and \(v^{*}\in L^{2}((0,T),H^{1}_{0}(\Omega ))\cap L^{\infty }(Q)\) such that
(see [26, Proposition 4.2]). Furthermore, by [7, Proposition 5.1] (5.3) holds true, and then we have
with \(v=v^{*}\), which leads to (5.4) being satisfied. □
Proposition 5.2
Assume that (1.1), \((R)\), and (G) are satisfied. Let \(u_{n}\) be the weak solution of problem \((P_{n})\). Then there exist a subsequence \(\{u_{n_{k}}\}\) and \(u\in L^{\infty }((0,T),\mathcal{M}^{+}(\Omega ))\) such that
Moreover, there exists a decreasing sequence \(\{A_{k}\}\subset Q\) of Lebesgue measurable sets with \(| A_{k}| \rightarrow 0\) as \(k\rightarrow \infty \) such that
where \(\tau \in \mathcal{Y}(Q,\mathbb{R})\) is the Young measure associated with \(\{u_{n_{k}}\}\) and
Proof
By (4.5) and Proposition 4.2, we apply the compactness theorem given by [27], then there exist \(u\in \mathcal{M}^{+}(Q)\) and a subsequence \(\{u_{n_{k}}\}\) such that \(u_{n_{k}}\overset{*}{\rightharpoonup }u \) in \(\mathcal{M}^{+}(Q)\). As argued in [1, 10, 28], we obtain \(u\in L^{\infty }((0,T),\mathcal{M}^{+}(\Omega ))\). Since (4.5) and the compactness result implies that \(\{u_{n_{k}}\}\) is bounded in \(L^{1}(Q)\). By Proposition 2.2, there exist a sequence of \(\{u_{n_{k}}\}\subseteq \{u_{n}\}\) and a Young measure \(\tau \in \mathcal{Y}(Q,\mathbb{R})\), and from Proposition 2.3 [Bitting Theorem], there exists a sequence of measure sets \(A_{k}\subseteq Q\), \(A_{k}\subseteq A_{k+1}\) and \(| A_{k}| \rightarrow 0\) such that
where \(u_{b}\in L^{1}(Q)\), \(u_{b}\geq 0\) is a barycenter of the limiting Young measure τ associated with the subsequence \(\{u_{n_{k}}\}\). Moreover, repeating the same proof, we show that \(\operatorname{supp} \tau _{(x,t)}\subseteq [0,+\infty )\) and \(\tau _{(x,t)}=\tau _{(x,t)}\llcorner [0,+\infty )\) for almost everywhere \((x,t)\in Q\), where \(\tau _{(x,t)}\) is the disintegration of the Young measure τ.
By (4.5) and the compactness result, the sequence \(\{u_{n_{k}}\chi _{Q\backslash A_{k}}\}\) is uniformly bounded in \(L^{1}(Q)\). Therefore, there exists a Radon measure \(\mu \in \mathcal{M}^{+}(Q)\) such that \(u_{n_{k}}\overset{*}{\rightharpoonup }\mu \) in \(\mathcal{M}^{+}(Q)\). Finally, the sequence \(u_{n_{k}}\) is of \(u_{n_{k}}=u_{n_{k}}\chi _{A_{k}}+u_{n_{k}}\chi _{Q\backslash A_{k}} \overset{*}{\rightharpoonup }\mu + u_{b}\) in \(\mathcal{M}^{+}(Q)\). Hence \(\mu :=u-u_{b}\) in \(\mathcal{M}^{+}(Q)\) holds true. □
Proposition 5.3
Let \(u_{n_{k}}\), \(u_{b}\in L^{\infty }((0,T),L^{1}(\Omega ))\) and \(\mu \in L^{\infty }((0,T),\mathcal{M}^{+}(\Omega ))\) be respectively the subsequence, function, and the measure given in Proposition 5.2. Then there exist a zero Lebesgue measure set \(\mathcal{N}\subset (0,T)\) and the subsequence (denoted again \(\{u_{n_{k}}\}\) such that, for any \(t\in (0,T)\backslash \mathcal{N}\), there holds
Moreover, we get
where \(u(\cdot ,t)=\mu (\cdot ,t)+ u_{b}(\cdot ,t)\) for any \(t\in (0,T)\backslash \mathcal{N}\).
Proof
This proof is similar to [1, Proposition 7.4], we argue this proof in two steps:
Step 1. Assume that \(h\in C^{2}(\mathbb{R}_{+})\) is such that
For every \(\rho \in C^{2}_{c}(\Omega )\), set
Let us prove that
where \(h^{*}\in L^{\infty }((0,T),L^{1}(\Omega ))\) is defined by
By (4.5), we obtain that
The purpose of this step is to prove \(U^{\rho }_{h,k}\in W^{1,1}(0,T)\) for every k.
In fact, the weak derivative of \(U^{\rho }_{h,k}\) is given by
Hence, there holds
Since \(h'\) is bounded and \(h''\) is compactly supported in \(\mathbb{R}_{+}\), by (4.8), (4.9), (4.11), and assumption (G), we may estimate each term of (5.15), then one has
It follows that
By assumption (G)-(iii) and (4.9), there exists a positive constant \(C_{\rho }=C(\rho )\) such that
On the other hand, we have
Accordingly, there exists a positive constant \(\widetilde{C}_{\rho }=\widetilde{C}(\rho )\) such that
In view of (5.15)–(5.18), the sequence \(\{U^{\rho }_{h,k}\}\) is uniformly bounded in \(W^{1,1}(0,T)\), whence relatively compact in \(L^{1}(0,T)\). In particular, there exists a subsequence \(\{U^{\rho }_{h,k_{j}}\}\) depending on ρ and h, where a function \(U^{\rho }_{h}\in L^{1}(0,T)\) is such that
Since we have
where \(\varphi \in C(\mathbb{R}_{+})\cap L^{\infty }(\mathbb{R}_{+})\) (see assumption \((H)\)). By Proposition 2.2, we have
where \(\varphi ^{*}\) is defined by
In particular, combining (5.21) with (5.8), one has
where \(u_{b}\in L^{\infty }((0,T),L^{1}(\Omega ))\) and \(\mu \in L^{\infty }((0,T),\mathcal{M}^{+}(\Omega ))\) are respectively the function and measure in Proposition 5.2, and so
Moreover, we obtain
whence by (5.19) there holds
for every \(t\in (0,T)\backslash \mathcal{N}\).
Step 2: Assume that, for every \(\rho \in C^{2}_{c}(\Omega )\), set
For h given in (5.20), we infer that
where \(U^{\rho }_{h,k}\) and \(U^{\rho }_{\varphi ,k}\) are defined in (5.11) and
Based on Proposition 5.2 with (5.25), we will show the following convergence:
where
From (5.20), there holds
for \(\text{a.e.} (x,t)\in Q\).
To prove (5.27), we consider for every \(\rho \in C_{c}(\Omega )\) and \(\{\rho _{k}\}\subseteq C^{2}_{c}(\Omega )\) be any sequence such that \(\rho _{k}\rightarrow \rho \) uniformly in Ω̅, then we get
By Step 1, one has
Therefore, we obtain
By letting \(j\rightarrow \infty \) in the above inequality, assertion (5.27) holds true. Therefore, for every ρ, there exist a subsequence denoted again by \(\{u_{n_{k}}\}\) and a zero Lebesgue measure set \(\mathcal{N}\subset (0,T)\) such that we get
for any \(t\in (0,T)\backslash \mathcal{N}\) and every \(\rho \in C_{c}(\Omega )\), hence (5.10) and (5.11). □
Proposition 5.4
Suppose that (1.1), (1.4), \((R)\), and (G) hold. Let μ be given by Proposition 5.2. Then there holds
and
where \(f^{*} ,\psi ^{*}\in L^{\infty }((0,T),L^{1}(\Omega ))\) is defined by
for a.e. in Q.
The proof of Proposition 5.4 is argued as in [14, Proposition 5.2] or [1, Lemma 8.3], for this reason we omit this proof.
Proposition 5.5
Assume that (1.1), (1.4), \((R)\), and (G) hold. Let μ be given by Proposition 5.2. Then there exist a zero Lebesgue measure set \(\mathcal{N}\subset (0,T)\) and a subsequence of \(\{u_{n_{k}}\}\) such that, for any \(t\in (0,T)\backslash \mathcal{N}\), there holds
and
Proof
Let \(\tau \in \mathcal{Y}(Q,\mathbb{R})\) and \(\{u_{n_{k}}\}\) be respectively the Young measure and the subsequence associated with the sequence of Young measure \(\{\tau ^{n}\}\). For every \(\rho \in C_{c}(\overline{\Omega })\),
To prove convergence (5.32), it is enough to show that
for every \(\xi \in C^{1}_{c}(0,T)\), with
and \(\mu (\cdot ,t)\in \mathcal{ M}^{+}(\Omega )\) and \(\psi ^{*}\in L^{\infty }(Q)\) given by (3.4). From assumption (G) and (1.1), the function \(\psi \in C^{2}(\mathbb{R}_{+})\), and for any \(\rho \in C^{1}_{c}(\Omega )\), for every \(k\in \mathbb{N}\), there holds
Moreover, let us consider \(\rho \psi '(u_{n_{k}})\xi \). For every \(\xi \in C^{1}_{c}(0,T)\) as a test function in \((P_{n})\), there holds
which leads to the fact that the weak derivation of \(G_{k}^{\rho }\in W^{1,1}(0,T)\) is such that
Since f, \(f'\), \(\psi '\), and \(\psi ''\) are continuous and uniformly bounded (see assumptions (G) and \((R)\)), then we have
By (4.8), (4.9), (4.11), and assumptions (G) and \((R)\), we may estimate each term of (5.39) as follows:
Therefore, there exists a positive constant \(C_{\rho }=C(\rho )\) such that
On the other hand, we have
Accordingly, there exists a positive constant \(\widetilde{C}_{\rho }=\widetilde{C}(\rho )\) such that
In view of (5.39)–(5.42), the sequence \(\{G^{\rho }_{h,k}\}\) is uniformly bounded in \(W^{1,1}(0,T)\), whence relatively compact in \(L^{1}(0,T)\). To achieve this proof, we consider for every \(\rho \in C_{c}(\Omega )\) and \(\{\rho _{k}\}\subseteq C^{2}_{c}(\Omega )\) be any sequence such that \(\rho _{k}\rightarrow \rho \) uniformly in Ω̅, then we get
By letting \(j\rightarrow \infty \) in the above inequality, assertion (5.32) holds true. We use a similar approach to prove convergence (5.33). Hence, we omit the proof of assertion (5.33). □
Remark 5.1
Since ψ is given in (1.1), assumption (H) and (5.8) hold, then there exists a subsequence of \(\{u_{n_{k}}\}\) (not relabeled) such that
where \(\varphi ^{*}\) is defined by (5.22) and u is given in Proposition 5.2. Similarly, we get
where \(t\in (0,T)\backslash \mathcal{N}\).
Proposition 5.6
Suppose that (1.1), (1.4), \((R)\), and (G) hold. Let (5.8) be the Lebesgue decomposition of u. Then there holds
Moreover,
where \(u_{r}\in L^{\infty }((0,T),L^{1}(\Omega ))\) is the density of an absolutely continuous part of u.
Proof
Let us recall the definition of the weak solution to problem \((P_{n})\) with \(m\in \mathbb{N}\backslash \{0\}\) such that
whenever \(\psi _{m}(\lambda )=\psi (\lambda )\chi _{(-m,\frac{1}{m})}(\lambda )\) and \(f_{m}(\lambda )=f(\lambda )\chi _{(-m,\frac{1}{m})}(\lambda )\). From Proposition 5.2 – Proposition 5.4, the following assertions hold true:
and
where
and
belong to \(L^{\infty }((0,T),L^{1}(\Omega ))\). Since we assume that ξ is a solution to the backward parabolic equations \((\nu \cdot \alpha )\) such that \(\xi _{\nu }\in C(Q)\), we get
and
Since, for all \(u\geq 0\), \(| \psi _{m}(u) | \leq (\alpha +\beta )u\chi _{(-m,\frac{1}{m})}(u)\) and \(| f_{m}(u) | \leq L u\chi _{(-m,\frac{1}{m})}(u)\), then \(| \psi _{m}^{*}(x,t) | \leq (\alpha +\beta )u_{b}^{m}\), where \(L:=\| f'(u) \| _{L^{\infty }(\mathbb{R}_{+})}\) and \(| f_{m}^{*}(x,t) | \leq L u_{b}^{m}\). Therefore, the following convergence holds: \(u_{b}^{m}\rightarrow 0\), \(\psi _{m}^{*}\rightarrow 0\), and \(f_{m}^{*}\rightarrow 0\) a.e. in Q as \(m\rightarrow \infty \). For all \(\overline{t}\in (0,T)\backslash \mathcal{N}\), with \(| \mathcal{N}| =0\). From (5.48)–(5.53), we deduce that
Taking the test function \(\xi \in C([0,T],C_{0}^{1}(\Omega ))\) such that \(\xi _{\nu }=0\) in \(\Omega \times (0,\overline{t})\). Hence \(\mu (\cdot ,t)\) is singular with respect to the Lebesgue measure and \(\mu (\cdot ,t)=[\mu (\cdot ,t)]_{s}=u_{s}(\cdot ,\overline{t})\) for any \(\overline{t}\in (0,T)\backslash \mathcal{N}\). □
6 Characterization of the limit Young measure
The main result of this section is given by the following proposition.
Proposition 6.1
Let \(v\in L^{2}((0,T),H^{1}_{0}(\Omega ))\cap L^{\infty }((0,T),H^{1}_{0}( \Omega ))\cap L^{\infty }(Q)\) be the limit of function given in Proposition 5.1. (i) Let \(\{\sigma _{n}\}\subseteq \mathcal{Y}(Q,\mathbb{R})\) be the sequence of Young measures on \(Q\times \mathbb{R}\) associated with \(\{ \psi (u_{n}) \} \). Then there exist Young measure \(\sigma \in \mathcal{Y}(Q,\mathbb{R})\) and a subsequence \(\{ \psi (u_{n_{k}}) \} \subseteq \{ \psi (u_{n}) \} \) such that
for almost everywhere \((x,t)\in Q\), there holds
where \(\sigma _{(x,t)}\in \mathcal{P}(\mathbb{R})\) is the disintegration of σ.
(ii) Let \(\{\tau _{n}\}\subseteq \mathcal{Y}(Q,\mathbb{R})\) be the sequence of Young measures on \(Q\times \mathbb{R}\) associated with \(\{u_{n}\}\). Then there exist Young measure \(\tau \in \mathcal{Y}(Q,\mathbb{R})\) and a subsequence \(\{u_{n_{k}}\}\subseteq \{u_{n}\}\) such that
Denoting by \(\tau _{(x,t)}\) the disintegration of τ, for almost everywhere \((x,t)\in Q\), there hold
where \(\delta _{ \{ \psi ^{-1}(v(x,t)) \} }\) is a Dirac mass concentration at \(\psi ^{-1}(v(x,t))\).
The proof of Proposition 6.1 is postponed now, and it will be made at the end of this section due to a number of intermediate steps: Proposition 6.2, Proposition 6.3, and Proposition 6.4.
Let us consider the function V as follows:
Proposition 6.2
Let V be the function (6.5), and let \(\tau _{(x,t)}\) be the disintegration of the limit Young measure τ mentioned in Proposition 5.2. Then there exist a subsequence \(\{u_{n_{k}}\}\subseteq \{u_{n}\}\) and a zero Lebesgue measure set \(\mathcal{N}\subset (0,T)\) such that
for every \(t\in (0,T)\backslash \mathcal{N}\).
Proof
Fix any \(\rho \in C^{1}_{c}(\Omega )\) and \(\epsilon >0\). Set
where
Let us first show that \(W^{\rho }_{\epsilon ,n}\in W^{1,1}(0,T)\).
By [28, Proposition 4.10], there exists an open set \(\Omega ^{t}_{n}\subseteq \Omega \) (depending on t) such that \(\operatorname{dist} ( \Omega ^{t}_{n},S^{t})>0\) and \(\operatorname{supp} \mathcal{T}_{\epsilon }(u_{n})( \cdot ,t)\subset \Omega ^{t}_{n}\) and \(u_{n}(\cdot ,t)\in H^{1}(\Omega ^{t}_{n})\), where \(S^{t}= \{ x\in \overline{\Omega }/\psi (u_{r})(x,t)\neq \infty \} \), then there holds
The weak derivative of \(W^{\rho }_{\epsilon ,n}\) is given by
for every \(t\in (0,T)\backslash \mathcal{N}\).
In view of (1.1), (4.5), (4.8), (6.8), and assumption (G), there exists a positive constant \(\widehat{C}_{\rho }=\widehat{C}(\rho )\) such that
Notice that estimate (6.12) holds for every \(\alpha \neq 0\).
Then sequence \(\{V_{\epsilon }(u_{n})\}\) is uniformly bounded in \(L^{1}(\Omega )\) and equi-integrable. In fact, for almost everywhere \((x,t)\in Q\), there holds
By (6.10) and (6.11), the family \(\{W^{\rho }_{\epsilon ,n}(t)\}\) is uniformly bounded in \(W^{1,1}(0,T)\). Therefore, there exist a subsequence \(\{u_{n_{k}}\}\subseteq \{u_{n}\}\) and a function \(W^{\rho }_{\epsilon }\in L^{1}(0,T)\) such that
where
Notice that the proof of (6.14) is argued as in Proposition 5.3. Therefore, for any \(\xi \in L^{\infty }(0,T)\), we have
By the dominated convergence theorem, there holds
whenever
for any \(t\in (0,T)\backslash \mathcal{N}\). □
Consider the orthogonal basic of \(L^{2}(\Omega )\) given by the operator −Δ with homogeneous Dirichlet conditions. Let \(\{\mu _{i}\}\) be the corresponding sequence of eigenvalues. Let \(P_{n}\), \(Q_{n}\): \(L^{2}(\Omega )\rightarrow H^{1}_{0}(\Omega )\), \(P_{n}+Q_{n}=I\) be the projection operator defined as follows:
for any \(\widetilde{f}\in L^{2}(\Omega )\).
Proposition 6.3
There exists \(C>0\) such that
We omit the proof of Proposition 6.3, the reader may refer to [28, Lemma 1], and this proposition is used in the proof of the following proposition.
Proposition 6.4
Let \(\{u_{n_{k}}\}\) be a subsequence mentioned in Proposition 6.2and ν̅ be the disintegration of the limit Young measure τ in Proposition 5.2. Let V be function (6.3), set also \(F:=\widetilde{f}\circ \psi \) with \(\widetilde{f}\in C(\mathbb{R})\) such that \(\| \widetilde{f}\| _{L^{\infty }(\mathbb{R})}+ \| \widetilde{f}'\| _{L^{\infty }(\mathbb{R})}\leq C\) for some constant \(C>0\). Then there holds
We omit the proof of Proposition 6.4, the reader should refer to [28, Proposition 6] for details. The importance of recalling this Proposition 6.4is that it can be proved for almost everywhere \((x,t)\in Q\), the disintegration measure \(\sigma _{(x,t)}\) is the Dirac mass concentrated at the point \(\psi ^{*}(x,t)\), where \(\psi ^{*}(x,t)\) is given by (5.31).
Now we can prove the main result of this section.
Proof of Proposition 6.1
(i) By Proposition 5.1, the function \(v\in L^{2}((0,T),H^{1}_{0}(\Omega ))\cap L^{\infty }((0,T),H^{1}_{0}( \Omega ))\cap L^{\infty }(Q)\) is such that (5.1), (5.2), and (5.4) hold. By statement (6.20), we choose the suitable functions F and V, then it can be proven for almost everywhere \((x,t)\in Q\), the disintegration σ (probably replacing \(\sigma _{(x,t)}\) by \(\tau _{(x,t)}\)) is a Dirac mass concentrated at \(\psi ^{*}(x,t)\) such that
The relevant proof is similar to that given in [41] and [42, Lemmas 5.1–5.2 and Theorem 2.12], then we omit it. To achieve the proof of the assertion (6.2), we should show that \(\psi ^{*}=v\) a.e. in Q. According to Proposition 5.6 and Proposition 5.2, we obtain
It follows that
(ii) Convergence (6.3) is a consequence of the Prokhorov theorem, since the sequence \(\{ u_{n}\}\) is uniformly bounded in \(L^{1}(Q)\) (see (4.5)). Furthermore, there exists a subsequence \(\{ u_{n_{k}}\}\subseteq \{ u_{n}\}\) such that (5.1)–(5.4) and (6.2) hold. Since \(\psi '>0\), then \(u_{n_{k}}\rightarrow \psi ^{-1}(v(x,t))\) a.e. in Q. Hence (6.4) is satisfied. □
Remark 6.1
(i) Assume that (1.1) and (G) are satisfied and \(v\in L^{2}((0,T),H^{1}_{0}(\Omega ))\cap L^{\infty }((0,T),H^{1}_{0}( \Omega ))\cap L^{\infty }(Q)\) as in Proposition 5.1, then \(\widetilde{S}=\{(x,t)\in \overline{Q}/ v(x,t)=\infty \}\) has a zero Lebesgue measure. Moreover,
(ii) \(u_{r}\in H^{1}(Q_{0})\) for any open subset \(Q_{0}\subseteq Q\) such that \(\operatorname{dist} (\overline{Q}_{0},\widetilde{S})>0\) and \(u_{r}\in C(\overline{Q}\backslash \widetilde{S})\), then
Proof of Theorem 3.1
The existence of solutions to problem (P) with qualitative properties follows from the statement of Proposition 5.2 – Proposition 5.6 and Proposition 6.1. □
7 Decay estimate of solutions
To establish the decay estimates, we make use of suitable test functions in the definition of weak solutions to the approximation problem (P) and the lower semi-continuity of the total variation theorem, then the estimate follows.
Proof of Theorem 3.2
Let us consider the approximation problem of \((Z_{n})\) given by
For every \(\epsilon >0\), we consider that \(\{\eta _{\epsilon }(t)\}\) is a sequence of smooth functions such that \(\| \eta _{\epsilon }(t)\| _{L^{1}(0,T)}\leq C\) and \(\eta _{\epsilon }(t)\overset{*}{\rightharpoonup }\delta _{t}\) in \(\mathcal{M}^{+}(0,T)\). Let us choose \(\xi (x,t)=\operatorname{sign}(u_{n}(x,t)-v_{n}(x,t))\int _{t}^{T}\eta _{ \epsilon }(s)s^{\alpha }\,ds\) as a test function in the approximation problem \((P_{n})-(Z_{n})\), then we have
By the integration by parts, the second term on the right-hand side vanishes, and we estimate the first term on the right-hand side, then we obtain
where \(C=C(T)>0\). By letting \(\epsilon \rightarrow 0^{+}\), we deduce that
By [9, Theorem 1, Sect. 5.2.1], the lower semi-continuity of the total variation, we get
for a.e. \(t\in (0,T)\) and \(\alpha >0\). Estimates (3.5) and (3.6) are achieved. □
Proof of Theorem 3.3
We argue this proof in two steps:
Step 1. We show that the solution of the nonlinear elliptic equation \((S)\) is Radon measure-valued.
To prove that the pseudo-stationary solutions w are Radon measure-valued solutions, we consider the approximation problem
where \(\{w_{n}\}\subseteq H^{1}_{0}(\Omega )\cap L^{\infty }(\Omega )\) and \(\{u_{0n}\}\subseteq C^{\infty }_{0}(\Omega )\) satisfies hypothesis (4.2). Let us consider the function \(\mathcal{T}_{\epsilon }\) for any \(\epsilon >0\) defined above such that
Notice that \(0\leq \mathcal{T}_{\epsilon }(s)\leq 1\) in \(\mathbb{R}_{+}\), \(\mathcal{T}_{\epsilon }(s)\rightarrow 1\) as \(\epsilon \rightarrow 0^{+}\). Taking \(\mathcal{T}_{\epsilon }(w_{n})\) as a function in \((S_{n})\), we obtain
By assumption (G), there holds
On the other hand, we consider \(T_{K}(\psi (w_{n}))=(1+\frac{1}{2}g_{K}(\psi (w_{n})))\psi (w_{n}) \chi _{(1,1+\epsilon )}(\psi (w_{n}))\) (see the proof of Proposition 4.2) is a test function in the approximation problem \((S_{n})\), then we obtain
Since \(1\leq T'_{K}(\psi (w_{n}))\leq 2\) and \(T_{K}(\psi (w_{n}))\in L^{\infty }(\mathbb{R}_{+})\), for every K, there holds
Since \(w_{n}\geq 0\) in Ω and assumption (4.2) holds, then we get
As argued in [18] and estimates (7.3) and (7.4), problem \((S)\) admits a Radon measure-valued solution i.e. \(w\in \mathcal{M}^{+}(\Omega )\).
Step 2. We show that convergence (3.8) holds true.
Let us consider \(\xi (x,t)=\operatorname{sign}(u_{n}(x,t)-w_{n}(x))\int _{t}^{T}\eta _{ \epsilon }(s)s^{\alpha }\,ds\) as a test function in the approximation problems \((S_{n})\) and \((P_{n})\), then we have
By the integration by parts, the second term on the right-hand side of the above equations vanishes, and we make use of estimate (7.3) and assumption (R). Then we can deduce that
By letting \(\epsilon \rightarrow 0^{+}\), therefore we obtain
where \(C=C(T)>0\). By virtue of [9, Theorem, Sect. 5.2.1], the lower semi-continuity of the total variation, we infer that
for a.e. \(t\in (0,T)\) and \(\alpha >0\). Since the constant C in estimate (7.6) is also uniform T, then by passing to the limit as \(t\rightarrow +\infty \) in the following inequality
convergence (3.8) is achieved. □
Availability of data and materials
Not applicable.
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Nkombo, Q.S., Li, F. & Tathy, C. Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations. Adv Differ Equ 2021, 509 (2021). https://doi.org/10.1186/s13662-021-03668-3
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DOI: https://doi.org/10.1186/s13662-021-03668-3