Blow-Up of Solutions for a Coupled Nonlinear Viscoelastic Equation with Degenerate Damping Terms: Without Kirchhoff Term

Salah Mahmoud Boulaaras,Mohamed Abdalla,Abdelbaki Choucha,Sahar Ahmed Idris,Karthikeyan Rajagopal,Sundarapandian Vaidyanathan

doi:10.1155/2021/6820219

Abstract

In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping and source terms without the Kirchhoff term. Under suitable hypothesis, we study the blow-up of solutions.

Highlights

In this paper, we consider the following problem:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ 􏼌􏼌􏼌􏼌ut􏼌􏼌􏼌􏼌ηutt − t Δu + 􏽚 h1(t −s)Δu(s)ds +􏼐|u|k +|v|l􏼑􏼌􏼌􏼌􏼌ut􏼌􏼌􏼌􏼌j−1ut f1(u, v), (x, t) ∈ Ω ×(0, T), 􏼌􏼌􏼌􏼌vt􏼌􏼌􏼌􏼌ηvtt − Δv + 􏽚 h2(t −s)Δv(s)ds +􏼐|v|θ + |u|〉􏼑􏼌􏼌􏼌􏼌vt􏼌􏼌􏼌􏼌s− 1vt⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
G1(u, ut) |ut|j− 2ut e author proved general and optimal decay of solutions. en, in [14], the author investigated the same problem without damping term and established a general decay of solutions
In problem (1) with η 0, in [15], Wu proved a general decay of solutions

Summary

Introduction

We consider the following problem:. f2(u, v), (x, t) ∈ Ω ×(0, T), u(x, t) v(x, t) 0, (x, t) ∈zΩ u(x, 0) u0(x), ut(x, 0) u1(x),. G1(u, ut) |ut|j− 2ut e author proved general and optimal decay of solutions. En, in [14], the author investigated the same problem without damping term and established a general decay of solutions. In problem (1) with η 0, in [15], Wu proved a general decay of solutions. In [16], Piskin and Ekinci established a general decay and blow-up of solutions with nonpositive initial energy for problem (1) case (Kirchhoff type). Some other authors investigate the hyperbolic type system with degenerate damping terms (see [17,18,19,20]). Under some restrictions on the initial datum and standard conditions on relaxation functions, the authors have established the global existence and proved the general decay of solutions. We give some concluding remarks in the last section

Preliminaries

Blow-Up