Study of weak solutions for parabolic variational inequalities with nonstandard growth conditions

In this paper, we study the degenerate parabolic variational inequality problem in a bounded domain. First, the weak solutions of the variational inequality are defined. Second, the existence and uniqueness of the solutions in the weak sense are proved by using the penalty method and the reduction method.


Introduction
This article concerned with initial-boundary problem whose model is with Lu = u t -div a(u)|∇u| p(x,t)-2 ∇uf (x, t), a(u) = u σ + d o , where ⊂ R + is a bounded simply connected domain, Q T = × (0, T], and T denotes the lateral boundary of the cylinder Q T . This type of variational inequality was studied initially by Chen and Yi [1], who proposed the equation V (x, 0) = g(x) i n (2) for modeling the American option. When r and σ are positive constant, the existence and uniqueness of solutions to problem (4) were also studied in [2][3][4].
In 2014, the authors in [5] discussed the problem with second-order elliptic operator They proved the existence and uniqueness of a solution to this problem with some conditions on u 0 , F, and L. Later, the authors in [6,7] extended the relative conclusions with the assumption that a(u) and p(x) are two positive constants. The author discussed the existence and uniqueness of a solution by the penalty method. The existence and uniqueness of such a problem with the assumption that p(x) and a(u) are variables were less studied.
The aim of this paper is to study the existence and uniqueness of solutions for a degenerate parabolic variational inequality problem. Throughout the paper, we assume that the exponent p(x, t) is continuous in Q = Q T with logarithmic module of continuity: where lim sup The outline of this paper is as follows. In Section 2, we introduce the function spaces of Orlicz-Sobolev type, give the definition of a weak solution to the problem, and prove the existence and uniqueness. Section 3 is devoted to the proof of the existence and uniqueness of the solution obtained in Section 2.

Basic spaces and the main results
To study our problems, let us introduce the Banach spaces: and denote by W (Q T ) the dual of W (Q T ) with respect to the inner product in L 2 (Q T ). In spirit of [3] and [4], we introduce the following maximal monotone graph: In addition, we define the following function class for the solution: , the following identity holds: The main theorem in this section is the following: (4). Suppose also that the following conditions hold: Then problem (1) has at least one weak solution in the sense of Definition 2.1.

Proof of the main results
In this section, we consider the family of auxiliary parabolic problems Here, M is a positive parameter to be chosen later. Moreover, and β ε (·) is the penalty function satisfying Following a similar method as in [6], we can prove that the regularized problem has a unique weak solution satisfying the following integral identities: and We start with two preliminary results that will be used several times.

Lemma 3.3 Let u ε be weak solutions of (5). Then
Proof First, we prove u ε ≥ u 0ε by contradiction. Assume that u ε ≤ u 0ε in Q 0 T , Q 0 T ⊂ Q T . Noting that u ε ≥ u 0ε on ∂Q T , we may assume that u ε = u 0ε on ∂Q 0 T . With (5) and letting t = 0, we deduce that From Lemma 3.2 we conclude that obtaining a contradiction. Second, we pay attention to u ε (t, x) ≤ |u 0 | ∞ + ε. Applying the definition of β ε (·), we have From (5) it is easy to prove that u ε (x, t) ≥ ε on ∂ × (0, T) and u 0ε (x) ≥ ε in . Thus, combining (21) and (23) and repeating Lemma 3.3, we have Third, we aim to prove (19). Since It follows by ε 1 ≤ ε 2 and the definition of β ε (·) that Thus, Lemma 3.3 can be proved by combining initial and boundary conditions in (5).
From [6] we can get the following inclusions: These conclusions, together with the uniform estimates in ε, allow us to extract from the sequence {u ε } a subsequence (for simplicity, we assume that it merely coincides with the whole sequence) such that for some functions u ∈ W (Q T ), A i (x, t) ∈ L p (x,t) (Q T ), and W i (x, t) ∈ L p (x,t) (Q T ).
Hence, (48) holds, and the proof of Lemma 3.10 is completed.
Applying (28), (29), and Lemma 3.10, it is clear that u(x, t) ≤ u 0 (x) in T , u(x, 0) = u 0 (x) in , ξ ∈ G(uu 0 ), and thus (a), (b), and (c) hold. The remaining arguments of the existence part are the same as those of Theorem 2.1 in [8], and we omit the details. Moreover, the uniqueness of solutions can be proved by repeating Lemma 3.1.