^{1}

^{1}

^{1}

We consider the following nonlinear parabolic equation:

Let

The study of partial differential equations involving variable-exponent nonlinearities has attracted the attention of researchers in recent years. The interest in studying such problems is stimulated and motivated by their applications in elastic mechanics, fluid dynamics, nonlinear elasticity, electrorheological fluids, and so forth. In particular, parabolic equations involving the

Regarding parabolic problems with nonlinearities of variable-exponent type, many works have appeared. We note here that most of the results deal with blow-up and global nonexistence. Let us mention some of these works. For instance, Alaoui et al. [

Recently, Shangerganesh et al. [

Equation

This paper consists of three sections in addition to the introduction. In Section

We present some preliminary facts about the Lebesgue and Sobolev spaces with variable exponents (see [

Let

Let

for

If (

We next define the variable-exponent Sobolev space

Let

If

Let

We end this section with a proposition which is exactly like Theorem

Let

In this section, we state and prove the well-posedness of our problem.

Let

We verify the conditions of Proposition

If

If

Hence

Hence,

To verify (

If

Therefore, conditions of Proposition

In this section, we present some numerical results and applications of the problem:

Assume that (

If

If

We consider two applications to illustrate numerically an exponential decay for the case

For this purpose, we introduce a numerical scheme for

In this part, we present a linearized numerical scheme to obtain the numerical results of the system

The parabolic equation

The semidiscrete problem is then written in a weak form to define the full-discrete problem: given

In this subsection, we present the following numerical applications of

Exponential decay: for

Polynomial decay: for

In both applications, we set the following parameters:

Figure

Mesh of the domain

The initial condition is taken to be

Initial condition:

The numerical results are obtained using the noncommercial software, FreeFem++ [

Numerical solutions for Application

With

Exponential decay.

Solid line:

This is also confirmed by Figure

The exponent function

Figure

Numerical solutions for Application

In this case, the solution

Polynomial decay.

Solid line:

We conclude that the numerical results in above applications verify Proposition

The authors declare that they have no conflicts of interest.

The authors acknowledge King Fahd University of Petroleum and Minerals for its support. This work is sponsored by KFUPM under Project no. FT 161004.