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This paper studies a time discretization for a doubly nonlinear parabolic equation related to the p(x)-Laplacian by using Euler-forward scheme. We investigate existence, uniqueness, and stability questions and prove existence of the global compact attractor.

The investigation of the asymptotic behavior for nonlinear parabolic equations involving the so-called p-Laplacian operator has been addressed by several authors in the last decades, in both bounded and unbounded domains, with constant or variable exponents (see [

In this paper, our goal is to study the time discretization for a doubly nonlinear parabolic equation associated with the p (x)-Laplacian, where in addition to usual questions of existence, uniqueness, and stability of the solutions, we will be concerned with the existence of absorbing sets and the global attractor as well. The problem under consideration is of the form

Existence results and qualitative properties concerning the solutions of the continuous problem (

Our motivation to study problem (

This paper is organized as follows: In Section

Finally, in Section

We begin with a review of some basic results that will be needed in the subsequent sections. The known results are stated without proofs. We shall however provide references where the proofs can be found.

We first introduce the space

Let

Let

The space

Let

Let

Let us recall the following versions of Poincaré’s inequality.

If

Let us next consider the modular version of Poincaré’s inequality.

Let

Let

We assume the following hypotheses:

Assume

To show that

Assume

We can write (

Let us now prove the uniqueness. For simplicity, we set

Then, applying

Assume

(i)

(ii)

(iii)

(i) From Lemma

(ii) In order to prove (

(iii) To prove point (

With the aid of the elementary identity,

For all

If

and

Assume

As

We can also derive a uniqueness result for problem (

Assume

Let

First case: suppose that

Second case: suppose that

Then, by using

Assume that

The proofs of existence and uniqueness are the same as those of Theorem

Now, we consider the following assumption:

Assume that

Since the proof is nearly the same as that of Theorem

The argument that allowed us to get (

As in (

In this section we consider the following problems: for all integer

We assume that

The result of Theorem

If

We multiply (

The following inequality holds, for any

By setting

Denote the left hand side of (

Now we are able to state our result on the existence of a compact attractor.

Suppose that

The nonlinear map

We define the

The authors declare that no data were used to support this study.

The authors declare that they have no conflicts of interest.