General Decay and Blow-Up of Solutions for a System of Viscoelastic Equations of Kirchhoff Type with Strong Damping

Wenjun Liu,Gang Li,Linghui Hong

doi:10.1155/2014/284809

Abstract

The general decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping is considered. We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy by exploiting the convexity technique, the other is for certain solutions with arbitrarily positive initial energy based on the method of Li and Tsai. Then, we give a decay result of global solutions by the perturbed energy method under a weaker assumption on the relaxation functions.

Highlights

We investigate the following system of viscoelastic equations of Kirchhoff type: t utt − M (‖∇u‖22) Δu + ∫ g1 (t − s) Δu (s) ds − Δut
To achieve general decay result we will use a Lyapunov type technique for some perturbation energy following the method introduced in [42]. This result improves the one in Li et al [23] in which only the exponential decay rates are considered
L ∗ (t) ≤ L ∗ (t0) e−k ∫tt0 ξ(s)ds, ∀t ≥ t0

Summary

Introduction

We investigate the following system of viscoelastic equations of Kirchhoff type:. Wu and Tsai [3] treated problem (2) for h(ut) = −Δut and proved the global existence, decay, and blow-up with suitable conditions on initial data They obtained the blow-up properties of local solution with small positive initial energy by the direct method of [4]. Many results concerning local existence, global existence, decay, and blow-up of solutions for a system of wave equations of Kirchhoff type without viscoelastic terms (i.e., gi = 0, i = 1, 2) have been extensively studied. Under suitable assumptions on the functions gi, fi (i = 1, 2), the initial data and the parameters in the above problem established local existence, global existence, and blow-up property (the initial energy E(0) < 0) This latter blow-up result has been improved by Messaoudi and Said-Houari [21] into certain solutions with positive initial energy.

Preliminaries and Main Result

Blow-Up of Solutions with Initial Data in the Unstable Set

Blow-Up of Solutions with Arbitrarily Positive Initial Energy

General Decay of Solutions