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Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping

Fei Liang12 and Hongjun Gao1*

Author Affiliations

1 Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, PR China

2 Department of Mathematics, Anhui Science and Technology University, Feng Yang 233100, Anhui, PR China

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Boundary Value Problems 2011, 2011:22  doi:10.1186/1687-2770-2011-22

Published: 13 September 2011

Abstract

In this paper, we consider the system of nonlinear viscoelastic equations

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with initial and Dirichlet boundary conditions. We prove that, under suitable assumptions on the functions gi, fi (i = 1, 2) and certain initial data in the stable set, the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, there are solutions with positive initial energy that blow up in finite time.

2000 Mathematics Subject Classifications: 35L05; 35L55; 35L70.

Keywords:
decay; blow-up; positive initial energy; viscoelastic wave equations