A Novel Decay Rate for a Coupled System of Nonlinear Viscoelastic Wave Equations

Khaled Zennir,Sultan S Alodhaibi

doi:10.3390/math8020203

Abstract

The main goal of the present paper is to study the existence, uniqueness and behavior of a solution for a coupled system of nonlinear viscoelastic wave equations with the presence of weak and strong damping terms. Owing to the Faedo-Galerkin method combined with the contraction mapping theorem, we established a local existence in [ 0 , T ] . The local solution was made global in time by using appropriate a priori energy estimates. The key to obtaining a novel decay rate is the convexity of the function χ , under the special condition of the initial energy E ( 0 ) . The condition of the weights of weak and strong damping has a fundamental role in the proof. The existence of both three different damping mechanisms and strong nonlinear sources make the paper very interesting from a mathematics point of view, especially when it comes to unbounded spaces such as R n .

Highlights

In this paper we investigate the coupled system
We proposed a more general nonlinearities in sources and used classical arguments (Holder, Young and Minkowski’s inequalities) to estimate them
The more complected case was considered, where we took a nonlinearity in the second derivative in time to get a more general case and obtain the first derivative in time for the variable in very large spaces (kut k Lκ (Rn ), kvt k Lκ (Rn ), θ θ this result is not classical)

Summary

Introduction

The more complected case was considered, where we took a nonlinearity in the second derivative in time (first term in both equations) to get a more general case and obtain the first derivative in time for the variable in very large spaces (kut k Lκ (Rn ) , kvt k Lκ (Rn ) , θ θ this result is not classical). These nonlinearities make the problem very interesting from the application point of view. We were obliged to show the global existence of unique solution

Preliminaries

Main Results

Proofs