Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations

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: {ut=αuxx+β[φ(u)]xx+f(u)inQ:=Ω×(0,T),u=0on∂Ω×(0,T),u(x,0)=u0(x)inΩ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} u_{t}=\alpha u_{xx}+\beta [\varphi (u) ]_{xx}+f(u) &\text{in} \ Q:=\Omega \times (0,T), \\ u=0 &\text{on} \ \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) &\text{in} \ \Omega , \end{cases} $$\end{document} where T>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T>0$\end{document}, Ω⊂R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega \subset \mathbb{R}$\end{document} is a bounded interval, u0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_{0}$\end{document} is nonnegative bounded Radon measure on Ω, and α,β≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha , \beta \geq 0$\end{document}, 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.

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 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 α = 0, problem (P) is nonlinear degenerate parabolic equations, and this kind of nonlinear degenerate parabolic equations is similar to is an open bounded domain with smooth boundary ∂ and u 0 is a finite Radon measure on . 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][45][46] are closely formulated as in (A.4), where the expression for the source term F(x, t, u) is more regular and the diffusionterm A(x, t, s) takes part on the modeling of real phenomena from mathematical biology and physics. Furthermore, the authors in [44][45][46] dealt with the properties of weak and classical solutions. Assuming that β = 0, problem (P) is reduced to the semilinear heat equation with Radon measure as initial data described as follows: where α ≡ 1, T > 0, ⊂ R is a bounded interval, u 0 is nonnegative bounded Radon measure on . 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 α > 0 and β > 0, and we notice that the result of this paper is not true for α = 0. The goal of this paper is threefold. Firstly, we study the existence of the Radon measurevalued 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 measurevalued 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. where |μ| stands for the total variation of μ.

Young measures
We denote by C c (R) the space of continuous real functionals with compact support in R and by M(R) the Banach space Radon measure on R endowed with the norm μ M(R) := |μ|(R) for any μ ∈ M(R).
By M(R) we denote the cone of positive finite Radon measure, and by P(R) the convex set of probability measure on R: τ M(R) := τ (R) for any τ ∈ P(R).
By a bounded Caratheodory integrand on A × R we mean that any function ϕ : A × R N → R is bounded and measurable, with ϕ(x, ·) continuous for almost everywhere x ∈ A. The duality map ·, · between the spaces M(R) and C c (R) is expressed as which can be extended to functions ρ ∈ C c (R). Let A ⊂ R N (N ≥ 1) be a bounded open set. We use the above equality to define the quantity μ, ρ R for any μ ∈ M(R) and every μ-integrable function ρ. Similar notation will be used for the space M(A × R) of finite Radon-measures on A × R. By Y(A, R) we denote the set of Young measures on A × R which are defined as follows (e.g. [15]).

Definition 2.1 A Young measure on
(2.5) If f ∈ L 1 (A), the Young measure associated with f is the measure τ ∈ Y(A, R) such that For any bounded Caratheodory integrand, there holds Let us recall the following result (e.g. [9,10,15]).
Then, for almost everywhere x ∈ A, there exists a probability measure τ x ∈ P(R) for any bounded Caratheodory integrand ϕ on A × R: is Lebesgue measurable; More generally, Proposition 2.1 holds true for very ϕ : A × R → R measurable and nonnegative or τ -integrable, we shall identify and τ ∈ Y(A, R) with the associated family {τ x |x ∈ A} which is called disintegration of τ .
for almost every x ∈ A and for any bounded Caratheodory integrand ϕ on A × R. Therefore, where δ p denotes the Dirac mass concentrated in p ∈ R.
The notion of narrow convergence of Young measures is as follows.
If τ n and τ are the Young measures associated with the measurable functions f n and f respectively, then τ n → τ narrowly if and only if f n → f in measure. In other words, f n → f in measure if and only if the Young measure associated with f n is δ f (x) (see [15,16]).
(ii) For > 0, there exists β > 0 such that, for any f ∈ U , Notice that for any function f ∈ 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 {τ n } be the sequence of associated Young measures. Then 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]).
and {f n k } be respectively the limiting Young measure and a subsequence given in Proposi- (2.14)

Statement of main results
Throughout this paper, we consider the backward parabolic equation which has a unique solution ξ in C 1,2 (Q) ∩ C 1 (Q) for any ξ ν ∈ C(Q) (see [12,13,30]), where where we denote by δ u r (x,t) the Dirac mass concentrated at u r (x, t) and The existence solution to problem (P) is given by the following result. Let us consider the following problem: In view of [12,13], problem (Z) has a unique solution function The estimate decay is given by the following result.
for any α > 0 and C is a positive constant. Moreover, if we extend t ∈ (0, T) into (0, +∞), 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 ∈ M + ( ) and the function ψ verifies hypothesis (G). Notice that the nonlinear elliptic equation (S) admits a nonnegative Radon measure-valued solution i.e. w ∈ M + ( ).
The main goal of the asymptotic behavior of the Radon measure-valued solutions to problem (P) is given in the following theorem. 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(·, t) ∈ M + ( ) for a.e. t ∈ (0, T). By the extension of the solution to global solutions on × (0, +∞), we infer that u(·, t) M + ( ) ≤ C for a.e. t ∈ (0, +∞) so that (3.8) is obtained.

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 ψ n is such that Since u 0 ∈ M + ( ), then the approximation of the Radon measure u 0 is given by [ for every ξ in C 1 (Q) such that ξ (·, T) = 0 and ξ = 0 on ∂ × (0, T). Now we establish some technical estimates which will be used in the proof of the existence solution. Proof Since ψ(u n ) = αu n + βϕ(u n ). Let us consider the nonlinear parabolic boundary value problem According to [ the normal outer derivative of ψ(u n )(x 0 , t 0 ) at ∂ × (0, T). Applying Green's formula for where 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 The sequence ψ(u n ) t is bounded in L 1 (0, T), H -1 ( ) + L 1 (Q). (4.10) Proof By the definition of ψ in (1.1), ψ(u n ) = αu n + βϕ(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.
For instance, let us consider {g K } ⊆ C 1 (R) such that g K (s) →sign(s) as K → +∞ for every s = 0 to be any sequence satisfying the following conditions: g K (0) = 0, |g K (s)| ≤ 1, g K (s) ≥ 0, |sg K (s)| ≤ 1 for every s ∈ R, and g K (s) = 0 if |s| ≥ 1 K . By recalling the sequence {g K } constructed in [1], we can construct the sequence of the function {T K } ⊆ C 1 (R + ) ∩ L ∞ (R + ) such that T K (s) = (1 + 1 2 g K (s))sχ (1,1+ ) (s). Assume that T K (ψ(u n )) is a test function to the approximation problem (P n ). Then we get Since 1 ≤ T K (ψ(u n )) ≤ 2 and T K (ψ(u n )), T K (ψ(s)) ∈ L ∞ (R + ) for every K , then (4.12) yields Then there exists a positive constant C = C( u 0 M + ( ) , f (u n ) L ∞ (R + ) , |Q|) > 0 such that Assume that ξψ (u n ) is a test function into the first equation of (P n ). Then we have for any ξ ∈ 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 (4.14) By hypothesis (G), ψ (u n ), ψ (u n ) ∈ L ∞ (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 {[ψ(u n )] t } in L 2 ((0, T), H -1 ( )) + 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 ∈ (0, T) such that t 1 + 1 < t 2 , and we observe that 0 ≤ h(s) < C, s ∈ (0, t 2 ). By assumption (G), it is easy to observe that for every n ∈ N [ψ(u n )] s = 0 on ∂ × (0, T) for the homogeneous Dirichlet condition. Multiplying the approximation problem (P n ) by the test function h(s)[ψ(u n )] s and integrating over × (0, t 2 ), we obtain It implies that In view of assumption (R), f (u n ) ∈ L ∞ (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.

Existence result
Now we study the limit points of the sequences {u n } and ψ(u n ) as n → ∞.
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 where τ ∈ Y(Q, 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 ∈ M + (Q) and a subsequence {u n k } such that u n k * u in M + (Q). As argued in [1,10,28], we obtain u ∈ L ∞ ((0, T), M + ( )). 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 } ⊆ {u n } and a Young measure τ ∈ Y(Q, R), and from Proposition 2.3 [Bitting Theorem], there exists a sequence of measure sets A k ⊆ Q, A k ⊆ A k+1 and |A k | → 0 such that where u b ∈ L 1 (Q), u b ≥ 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 supp τ (x,t) ⊆ [0, +∞) and τ (x,t) = τ (x,t) [0, +∞) for almost everywhere (x, t) ∈ Q, where τ (x,t) is the disintegration of the Young measure τ . By (4.5) and the compactness result, the sequence {u n k χ Q\A k } is uniformly bounded in L 1 (Q). Therefore, there exists a Radon measure μ ∈ M + (Q) such that u n k * μ in M + (Q).
Proof This proof is similar to [1,Proposition 7.4], we argue this proof in two steps: Step Let us prove that By (4.5), we obtain that 14) The purpose of this step is to prove U ρ h,k ∈ W 1,1 (0, T) for every k. In fact, the weak derivative of U ρ h,k is given by Hence, there holds Since h is bounded and h is compactly supported in 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 On the other hand, we have Accordingly, there exists a positive constant In view of (5.15)-(5.18), the sequence {U ρ 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 ρ h,k j } depending on ρ and h, where a function U ρ h ∈ L 1 (0, T) is such that Since we have where ϕ ∈ C(R + ) ∩ L ∞ (R + ) (see assumption (H)). By Proposition 2.2, we have where ϕ * is defined by In particular, combining (5.21) with (5.8), one has h(u n k ) = u n kϕ(u n k ) * where u b ∈ L ∞ ((0, T), L 1 ( )) and μ ∈ L ∞ ((0, T), M + ( )) are respectively the function and measure in Proposition 5.2, and so Moreover, we obtain for every t ∈ (0, T)\N .
To prove (5.27), we consider for every ρ ∈ C c ( ) and {ρ k } ⊆ C 2 c ( ) be any sequence such that ρ k → ρ uniformly in , then we get By Step 1, one has Therefore, we obtain By (4.5) and (5.28), one has lim sup By letting j → ∞ 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 N ⊂ (0, T) such that we get for any t ∈ (0, T)\N and every ρ ∈ C c ( ), hence (5.10) and (5.11). where f * , ψ * ∈ L ∞ ((0, T), L 1 ( )) 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.
Proof Let τ ∈ Y(Q, R) and {u n k } be respectively the Young measure and the subsequence associated with the sequence of Young measure {τ n }. For every ρ ∈ C c ( ), To prove convergence (5.32), it is enough to show that and μ(·, t) ∈ M + ( ) and ψ * ∈ L ∞ (Q) given by (3.4). From assumption (G) and (1.1), the function ψ ∈ C 2 (R + ), and for any ρ ∈ C 1 c ( ), for every k ∈ N, there holds Moreover, let us consider ρψ (u n k )ξ . For every ξ ∈ C 1 c (0, T) as a test function in (P n ), there holds By letting j → ∞ 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).

Characterization of the limit Young measure
The main result of this section is given by the following proposition. Denoting by τ (x,t) the disintegration of τ , for almost everywhere (x, t) ∈ Q, there hold , 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: (6.5) Proposition 6.2 Let V be the function (6.5), and let τ (x,t) be the disintegration of the limit Young measure τ mentioned in Proposition 5.2. Then there exist a subsequence {u n k } ⊆ {u n } and a zero Lebesgue measure set N ⊂ (0, T) such that for every t ∈ (0, T)\N .
Notice that the proof of (6.14) is argued as in Proposition 5.3. Therefore, for any ξ ∈ L ∞ (0, T), we have By the dominated convergence theorem, there holds for any t ∈ (0, T)\N .
Consider the orthogonal basic of L 2 ( ) given by the operator -with homogeneous Dirichlet conditions. Let {μ i } be the corresponding sequence of eigenvalues. Let P n , Q n : L 2 ( ) → H 1 0 ( ), P n + Q n = I be the projection operator defined as follows: for any f ∈ L 2 ( ).

Proposition 6.3
There exists C > 0 such that P n ψ(u n ) L 2 ((0,T),H 1 0 ( )) + n 1 2 P n ψ(u n ) L 2 (Q) ≤ C. (6.19) 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.
We omit the proof of Proposition 6.4, the reader should refer to [28,Proposition 6] for details. The importance of recalling this Proposition 6.4 is that it can be proved for almost everywhere (x, t) ∈ Q, the disintegration measure σ (x,t) is the Dirac mass concentrated at the point ψ * (x, t), where ψ * (x, t) is given by (5.31).

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. 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 T 0 u n (x, t)v n (x, t) η (t)t α dx dt ≤ C u 0 M + ( ) , (7.1) where C = C(T) > 0. By letting → 0 + , we deduce that 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 measurevalued.
To prove that the pseudo-stationary solutions w are Radon measure-valued solutions, we consider the approximation problem Notice that 0 ≤ T (s) ≤ 1 in R + , T (s) → 1 as → 0 + . Taking T (w n ) as a function in (S n ), we obtain ψ (w n )T (w n )|w nx | 2 dx + T (w n )w n dx ≤ u 0 M + ( ) .
Step 2. We show that convergence (3.8) holds true. Let us consider ξ (x, t) = sign(u n (x, t)w n (x)) T t η (s)s α ds as a test function in the approximation problems (S n ) and (P n ), then we have