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

Khaled Zennir and Sultan S. Alodhaibi
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Khaled Zennir: Department of Mathematics, College of Sciences and Arts at Ar Rass, Qassim University, Ar Rass 51921, Saudi Arabia
Sultan S. Alodhaibi: Department of Mathematics, College of Sciences and Arts at Ar Rass, Qassim University, Ar Rass 51921, Saudi Arabia

Mathematics, 2020, vol. 8, issue 2, 1-12

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 .

Keywords: viscoelastic wave equation; coupled system; global solution; Faedo–Galerkin approximation; decay rate (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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