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July  2004, 10(3): 787-804. doi: 10.3934/dcds.2004.10.787

Qualitative analysis of a nonlinear wave equation

Jorge A. Esquivel-Avila 1,

1. 

Departamento de Ciencias Básicas, Análisis Matemático y sus Aplicaciones, UAM-Azcapotzalco, Av. San Pablo 180, Col. Reynosa Tamaulipas, México , D. F., 02200, Mexico

Received  July 2002 Revised  July 2003 Published  January 2004

We study the qualitative behavior of solutions of a wave equation with nonlinear damping and a source term. We give a characterization of blow-up of solutions, improving a previous result. When the dissipation dominates the source term, we show existence of unbounded global solutions. We use the stable (potential well) and unstable sets, introduced by Sattinger and Payne. We study all bounded global solutions, and we charaterize their convergence as $t \rightarrow \infty$. In particular, we prove that every solution, with energy larger or equal than the depth of the potential well, is global, bounded and converges to the set of nonzero equilibria.
Keywords: convergence to equilibria, source term, decay rates., blow-up, global and unbounded solutions, nonlinear dissipation, Wave equation.
Mathematics Subject Classification: 35L70, 35B3.
Citation: Jorge A. Esquivel-Avila. Qualitative analysis of a nonlinear wave equation. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 787-804. doi: 10.3934/dcds.2004.10.787
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