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

Department of Mathematics, College of Sciences and Arts at Ar Rass, Qassim University, Ar Rass 51921, Saudi Arabia
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Mathematics 2020, 8(2), 203; https://doi.org/10.3390/math8020203
Received: 19 December 2019 / Revised: 1 February 2020 / Accepted: 2 February 2020 / Published: 6 February 2020
(This article belongs to the Special Issue Mathematical Methods in Nonlinear Waves and Dynamical Systems)
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 viscoelastic wave equation; coupled system; global solution; Faedo–Galerkin approximation; decay rate
MDPI and ACS Style

Zennir, K.; Alodhaibi, S.S. A Novel Decay Rate for a Coupled System of Nonlinear Viscoelastic Wave Equations. Mathematics 2020, 8, 203.

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