WELL-POSEDNESS AND LARGE-TIME BEHAVIORS OF SOLUTIONS FOR A PARABOLIC EQUATION INVOLVING p(x)-LAPLACIAN

This paper is concerned with the initial-boundary value problem for a nonlinear parabolic equation involving the so-called p(x)-Laplacian. A subdifferential approach is employed to obtain a well-posedness result as well as to investigate large-time behaviors of solutions.


Introduction.
Let Ω ⊂ R N be a bounded domain with smooth boundary ∂Ω. The so-called p(x)-Laplacian is given by with a function p = p(x) from Ω into (1, ∞). In the present paper we study the well-posedness and asymptotic behaviors of solutions u = u(x, t) as t → ∞ for the following initial-boundary value problem: where ∂ t u = ∂u/∂t and f : Ω × (0, ∞) → R and u 0 : Ω → R are given functions. The p(x)-Laplacian with a variable exponent p(x) is deeply related to generalized Lebesgue and Sobolev spaces, L p(x) and W 1,p(x) . There have been many contributions to nonlinear elliptic problems associated with the p(x)-Laplacian (see, e.g., [23] for a thorough overview of the recent advantages) from various view points. Moreover, parabolic equations involving the p(x)-Laplacian have been proposed in the study of image restoration (see [13]) as well as in some model of electrorheological fluids (see [14], [15], [26]). A mathematical analysis was also done for such problems by Acerbi and Mingione [1,2] and by Acerbi, Mingione and Seregin [3]. In [22], some nonlinear parabolic problem proposed by [13] was studied in a weak formulation and an existence result for weak solutions was established. Antontsev and Shmarev studied parabolic equations with anisotropic p(x, t)-Laplace operators and proved existence, uniqueness, extinction in finite time, decay and blow-up of solutions in [5,6,7,8].
Equation (1) was very recently studied by Bendahmane, Wittbold and Zimmermann in [10], where the well-posedness of a renormalized solution is proved for L 1 -data. Moreover, the existence and uniqueness of entropy solutions and the equivalence between two notions of solutions are also discussed by Zhang and Zhou in [28]. In this paper we treat L 2 -solutions for (1)-(3), and we prove the well-posedness and reveal large-time behaviors of solutions by using subdifferential calculus.
This paper is composed of four sections. In Section 2, we recall the definition of variable exponent Lebesgue spaces, L p(x) (Ω), as well as Sobolev spaces, W 1,p(x) (Ω). Moreover, some properties of these spaces will be also exhibited to be used later. In Section 3, we prove the well-posedness of the Cauchy-Dirichlet problem (1)-(3) by using a theory of evolution equations governed by subdifferential operators. Moreover, we also treat the periodic problem for (1). In Section 4, we discuss asymptotic behaviors of solutions u = u(x, t) for (1)-(3) as t → ∞.
Further results on qualitative properties of solutions for (1)-(3) (e.g., extinction/decay rates of solutions and limit problems as p(x) → ∞) will be reported in our forthcoming paper [4].

2.
Generalized Lebesgue and Sobolev spaces. This section is devoted to some preliminary facts on Lebesgue and Sobolev spaces with variable exponents (see [24], [16,17], [20] for an introduction to this field). Let Ω be a bounded domain in R N . Let p be a measurable function from Ω to [1, ∞). We write Define a Lebesgue space with a variable exponent p(x), which is a special sort of Musielak-Orlicz spaces (see [25]), by The following proposition plays an important role to establish energy estimates (see Theorem 1.3 of [20] for a proof).

Proposition 1. It holds that
with the strictly increasing functions, We next define variable exponent Sobolev spaces W 1,p(x) (Ω) as follows: The following proposition is concerned with the uniform convexity of L p(x) and W 1,p(x) (see [25] for its proof).
Let us exhibit Poincaré and Sobolev inequalities (see [18], [21], [27] and references therein for more details). To do so, we introduce the Zhikov-Fan condition: with some A, δ > 0. This condition follows from a Hölder continuity of p over Ω.

Proposition 3.
Let Ω be a bounded domain in R N with smooth boundary ∂Ω.
In particular, the space W 1,p(x) 0 (Ω) has a norm · 1,p(x) given by 3. Well-posedness. In this section, we prove the well-posedness of (1)-(3) by using a theory of evolution equations governed by subdifferential operators. Let us begin with the definition of solutions for (1)-(3).
One can verify the following proposition (see [19]).
Proof. Since p(x) ≥ 2 for a.e. x ∈ Ω, by the following fundamental inequality (see, e.g., [12]): which is used in a proof of Clarkson's first inequality, we derive for a.e. x ∈ Ω, which gives On the other hand, since w n → w weakly in L 2 (Ω) and ϕ is weakly lower-semicontinuous in L 2 (Ω), we observe that Combining these facts with the assumption that ϕ(w n ) → ϕ(w), we deduce that lim inf n→∞ ϕ w n − w 2 = 0, (Ω). It completes our proof.
Let us next discuss the existence and the uniqueness of periodic solutions.
Remark 2. In the proof described above, the Sobolev inequality (5) (i.e., the continuous embedding W 1,p(x) (Ω) ֒→ L 2 (Ω)) is used, but the compactness of the embedding is not employed. Hence the continuous embedding is sufficient for this proof, so one can replace (4) and (9) by the following: N + 2 ≤ p − , p + < ∞ and p is Lipschitz continuous on Ω.