The Existence Result of Renormalized Solution for Nonlinear Parabolic System with Variable Exponent and L1 Data

Fairouz Souilah ,Messaoud Maouni ,Kamel Slimani

doi:10.28924/2291-8639-18-2020-748

Abstract

In this paper, we prove the existence result of a renormalized solution to a class of nonlinear parabolic systems, which has a variable exponent Laplacian term and a Leary lions operator with data belong to L1.

Highlights

Let Ω is bounded open domain of RN, (N ≥ 2) with lipschiz boundary ∂Ω, T is a positive number oure aime is to study the existence of renormalized solution for a class of nonlinear parabolic systeme with variable exponent and L1 data
We prove the existence result of a renormalized solution to a class of nonlinear parabolic systems, which has a variable exponent Laplacian term and a Leary lions operator with data belong to L1
El Hamdaoui and all in [11] studied the renormalized solutions for nonlinear parabolic systems in the Lebesgue Sobolev Space with variable exponent and L1 data

Summary

Introduction

Let Ω is bounded open domain of RN , (N ≥ 2) with lipschiz boundary ∂Ω, T is a positive number oure aime is to study the existence of renormalized solution for a class of nonlinear parabolic systeme with variable exponent and L1 data. Nonlinear parabolic systems; variable exponent; renormalized solutions; L1 data. El Hamdaoui and all in [11] studied the renormalized solutions for nonlinear parabolic systems in the Lebesgue Sobolev Space with variable exponent and L1 data. In 2016 [17] authors proved the existence and uniqueness of renormalized solution of a reaction diffusion systems which has a variable exponent Laplacian term and could be applied to image denoising for the case of parabolic equations. Wittbold and A.Zimmermann [7] have proved the existence and uniqueness of renormalized solution to nonlinear parabolic equations with variable exponent and L1 data. Zhou studied the renormalized and entropy solution for nonlinear parabolic equation with variable exponent and L1 data. In the present paper we prove the existence of renormalized solution for nonlinear parabolic systems with variable exponent and L1 data of (1.1).

The Mathematical Preliminaries on Variable Exponent Sobolev Spaces

The Assumptions on The Data

The Main Results