Decay of solutions of the wave equation with localized nonlinear damping and trapped rays

Kim Dang Phung

doi:10.3934/mcrf.2011.1.251

Article Contents

2011, Volume 1, Issue 2: 251-265. Doi: 10.3934/mcrf.2011.1.251

This issue Previous Article Strict Lyapunov functions for semilinear parabolic partial differential equations Next Article

Decay of solutions of the wave equation with localized nonlinear damping and trapped rays

Kim Dang Phung^1,

1.
Yangtze Center of Mathematics, Sichuan University, Chengdu 610064

Received: November 2010

Revised: April 2011

Published: June 2011

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Abstract

We prove some decay estimates of the energy of the wave equation governed by localized nonlinear dissipations in a bounded domain in which trapped rays may occur. The approach is based on a comparison with the linear damped wave equation and an interpolation argument. Our result extends to the nonlinear damped wave equation the well-known optimal logarithmic decay rate for the linear damped wave equation with regular initial data.

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Mathematics Subject Classification: Primary: 35L05, 35L71; Secondary: 35B40, 35B35.

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