Parabolic inequalities in Orlicz spaces with data in L 1

: In this paper, we provide existence and uniqueness of entropy solutions to a general nonlinear parabolic problem on a general convex set with merely integrable data and in the setting of Orlicz spaces.


Introduction
In this paper, we deal with the boundary value problems where c x t , ( ) belongs to E Q c , 0 M ( ) ≥ and k i 1, 2 i ( ) = to + and α to * + .
There exists a real γ 1 > such that is a given closed convex subset of Q ( ) . Q ( ) is the space of measure with the finite total mass.

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The problem P ( ) has several applications in engineering, game theory, finance, and economics. For example, one of the most important problems in finance is the optimal investment problem of a constant relative risk aversion. This problem leads to an obstacle parabolic problem with free boundaries (see [1]). Other important cases are the obstacle problem for parabolic minimizers studied in [2], where the obstacle is irregular, the pricing of American options (see [3]), as well as the models of pricing a double defaultable interest rate swap for which the solutions converge to a solution of a PDE coupled with two-obstacle problem.
On a physical area, the use of PDE in the convex set takes considerable importance, for example, the Boltzmann equation in a strictly convex domain with the specular, bounce-back, and diffuse (see [4]), dissipation inequalities for nonlinear PDEs which can be applied according to the choice of the so-called supply rate [5], Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP) equation, etc.
It is well known that P ( ) admits at least one solution (see Leray and Lions [6], Browder and Brézis [7], and Puel [8]). In those papers, the function a x t ξ , , ( ) was assumed to satisfy a polynomial growth condition with respect to u ∇ . When trying to generalize the last condition of a ξ ., ( ) to the non polynomial one, we are led to replace the space L T W 0, ; Ω built from an Orlicz space L M instead of L p , where the N -function M, which defines L M , defines the new growth of the operator. Such type of extension of the growth condition is more realistic and appears in several physical phenomena.
Partial differential equations with data-only integrable received special attention. The cornerstone of the theory was initially developed by DiPerna and Lions [9], where they introduced the notion of renormalized solution for the Boltzmann equation. It was also developed by Boccardo et al. [10] and Murat [11]. Other notions of solutions solving the problem with L 1 -data are SOLA (solutions obtained as a limit of approximation), from the study by Boccardo and Gallouët [12,13]. Finally, entropy solutions are introduced by Benilan et al. [14], Boccardo et al. [15].
Our purpose in this paper is to prove existence results and uniqueness, of the entropy solution, of the problem P ( ) in the setting of the inhomogeneous Sobolev space W L . In these types of problems the proofs are essentially based on the good choice of the test functions. The new idea of this paper is considering the problem in nonreflexive Banach space, namely, the inhomogeneous Orlicz-Sobolev space, and in a general convex set.
The simplest model of our problem P ( ) is the case A u a x t u div ,

Preliminaries
Let us recall the following definitions of spaces and topologies that will be used later (for the detail, we refer the reader to the rich literature in [16][17][18][19]).

Inhomogeneous Orlicz-Sobolev spaces
The inhomogeneous Orlicz-Sobolev spaces are defined as follows: W L Q u L Q D u L Q : and equipped with the usual quotient norm. We also denote

Main results
The following lemmas will be of interest in the proof of our main results. Let us denote NC , then we have Proof. Let u be a continuous function, we say that u satisfies β ( ) condition if there exists a continuous and increasing function β such that u t u s β u t s 2 02 Let us consider the set v X v t C t T : , It is easy to see that is a closed convex (since all its elements satisfy β ( ) condition). □ We claim that the problem has a weak solution, which is unique in the sense defined in [21]. Indeed, let us consider the approximate problem: The existence of such u X n ∈ was ensured by Kacur [22]. Following the same proof as in [21], we can prove the existence of a solution u of the problem P ( ) ′ as limit of u n .
Theorem 3.1. Under hypotheses (1.1)-(1.6), the problem P ( ) has at least one entropy solution in the following sense: Proof. Let us define the indicator functional, Φ is weakly lower semicontinuous.

I. A priori estimate
Let us consider the following approximate problem: Then, there exist a subsequence (also denoted u n ( )) and a measurable function u such that for Π , Π , strongly in and a.e. in .
Coming back to the inequality (3.2), we have nT ( ) ∫ ≤ , and by letting k to infinity and using Fatou lemma, one has Suppose there exists a subsequence u n ( ) such that u n ∉ for all n, then which is a contradiction. Then, there exists a subsequence that we denote as also u n ( ) such that u n ∈ for all n.
In what follows, we only consider this subsequence.
To prove that u ∈ , we need to prove that u Q ( ) ∈ . For this reason, let us consider φ as the truncation defined by as a test function in the approximate problem P n ( ), we obtain n n ( ( )) = By using Lemma 3.1, we obtained by following the same way as in [20], we have for a good N -function D , and F t μ θ , . By using [24] (see Proposition 2.3), it follows that Then, u BV Q u L Q u Q :

II. Almost everywhere convergence of the gradients
The main tool in this step proves which gives by the same argument as in [25] and adapted to the parabolic case, u u n ∇ → ∇ a.e. in Q. This is possible by using the following regularization principle ω and ω μ is the mollifier function defined by Landes [26], ω x t , , where ϖ is the zero extension of ω for s T > . The function ω μ j i , have the following properties: in for the modular convergence with respect to , in for the modular convergence with respect to .
We will be interested to estimate the elements of the aforementioned equation.
, there exists a smooth function u nσ (see [23]) such that, for the modular convergence in and for the modular convergence in .  It is easy to remark that T u 0 k nσ For the first term and as for I   We will now treat the terms (3.8)- (3.9). Before that, we will give some convergence results.

V. Uniqueness
Following the same way as Theorem 5.1 [14] for the parabolic case, we obtain the uniqueness.

Conclusion
In this paper, we have focused on the existence, uniqueness, and regularity of a class of inequalities in a general convex set and in a nonstandard functional framework, which is the Sobolev Orlicz spaces. The techniques used are not standard and require a very particular handling of the test functions and the approximated problems.