On the strongly damped wave equation with nonlinear damping and source terms

Shengqi Yu

doi:10.14232/ejqtde.2009.1.39

Abstract

We consider a wave equation in a bounded domain with nonlinear dissipation and nonlinear source term. Characterizations with respect to qualitative properties of the solution: globality, boundedness, blow-up, convergence up to a subsequence towards the equilibria and exponential stability are given in this article.

Highlights

Let Ω ⊆ RN (N ≥ 1) be a bounded domain with smooth boundary ∂Ω
Cazenave [5] proved the boundedness of global solutions to (1.1) for ω = μ = 0, while EsquivelAvila [7] recovered the same result for ω = 0 and μ > 0 and showed that this property may fail in presence of nonlinear disspation, by exploiting the same technique in [7], we proved, under the restrictions E(t) ≥ d, ∀ t ≥ 0 and m < p, the global solutions can still be bounded even in presence of nonlinear weak damping
In the recent paper [4], thanks to a new combination of the potential well and concavity methods, the global nonexistence of solutions has been proved for Kirchhoff systems when ω = 0 and the initial energy is possibly above the critical level d

Summary

Introduction

Let Ω ⊆ RN (N ≥ 1) be a bounded domain with smooth boundary ∂Ω. We are concerned with the behavior of the following superlinear wave equation with dissipation. For equations with linear weak damping, we refer to [7, 10, 14]. Gazzola and Squassina [8] extended this result to the case when μ = 0 and E(0) ≤ d All those works mentioned above dealt with the linear damping case (m = 1) or when the weak damping is absent(μ = 0). In the recent paper [4], thanks to a new combination of the potential well and concavity methods, the global nonexistence of solutions has been proved for Kirchhoff systems when ω = 0 and the initial energy is possibly above the critical level d.

Preliminaries

Boundedness of Global Solutions

Global Existence

Blow up

Exponential Decay