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

Fei Liang,Hongjun Gao

doi:10.1186/1687-2770-2011-22

Abstract

Abstract In this paper, we consider the system of nonlinear viscoelastic equations u t t - Δ u + ∫ 0 t g 1 ( t - τ ) Δ u ( τ ) d τ - Δ u t = f 1 ( u , v ) , ( x , t ) ∈ Ω × ( 0 , T ) , v t t - Δ v + ∫ 0 t g 2 ( t - τ ) Δ v ( τ ) d τ - Δ v t = f 2 ( u , v ) , ( x , t ) ∈ Ω × ( 0 , T ) with initial and Dirichlet boundary conditions. We prove that, under suitable assumptions on the functions g i , f i (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.

Highlights

In this article, we study the following system of viscoelastic equations: ⎧ ⎪⎪⎪⎪⎨ utt vtt − − u+ v+ 00ttgg21((tt− τ) − τ) u(τ )dτ − v(τ )dτ −ut = f1(u, v), (x, t) ∈ vt = f2(u, v), (x, t) ∈× (0, T), × (0, T), ⎪⎪⎪⎪⎩
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
We study the following system of viscoelastic equations:

Summary

Introduction

We study the following system of viscoelastic equations: They established both exponential and polynomial decay results under the conditions on g and its derivatives up to the third order, whereas Berrimi and Messaoudi [6] allowed the internal dissipation to be nonlinear.

Results

Conclusion