Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping

Mohammad  A. Rammaha; Daniel Toundykov; Zahava Wilstein

doi:10.3934/dcds.2012.32.4361

Article Contents

2012, Volume 32, Issue 12: 4361-4390. Doi: 10.3934/dcds.2012.32.4361

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Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping

1.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588-0130
2.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588
3.
Department of Mathematics & Computer Science, Berry College, Mount Berry, GA 30149-5014

Received: May 31, 2011

Revised: December 31, 2011

Published: November 2012

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Abstract

This paper presents a study of the nonlinear wave equation with $p$-Laplacian damping: \[ u_{tt} - \Delta u - \Delta _p u_t = f(u) \] evolving in a bounded domain $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions. The nonlinearity $f(u)$ represents a strong source which is allowed to have a supercritical exponent, i.e., the Nemytski operator $f(u)$ is not locally Lipschitz from $H{1\atop 0}(\Omega)$ into $L^2(\Omega)$. The nonlinear term $- \Delta _p u_t $ acts as a strong damping where the $-\Delta _p$ denotes the $p$-Laplacian. Under suitable assumptions on the parameters and with careful analysis involving the Nehari Manifold, we prove the existence of a global solution and estimate the decay rates of the energy.

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Mathematics Subject Classification: Primary: 35L05, 35J92; Secondary: 35A01, 35D30, 35B35.

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