Higher order energy decay rates for damped wave      equations with variable coefficients

Petronela Radu; Grozdena Todorova; Borislav Yordanov

doi:10.3934/dcdss.2009.2.609

Article Contents

2009, Volume 2, Issue 3: 609-629. Doi: 10.3934/dcdss.2009.2.609

This issue Previous Article On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms Next Article Interface conditions for a singular reaction-diffusion system

Higher order energy decay rates for damped wave equations with variable coefficients

1.
Department of Mathematics, University of Nebraska-Lincoln, Avery Hall 239, Lincoln, NE 68588
2.
Department of Mathematics, University of Tennessee, Knoxville, TN 37096-1300
3.
Department of Mathematics, University of Tennessee-Knoxville, TN 37996

Received: September 30, 2008

Revised: January 31, 2009

Published: August 2009

Abstract Related Papers Cited by

Abstract

Under appropriate assumptions the energy of wave equations with damping and variable coefficients $c(x)$$u_{t t}$-div$(b(x)\nabla u)+a(x)u_t =h(x,t)$ has been shown to decay. Determining the decay rate for the higher order energies of the $k$th order spatial and time derivatives has been an open problem with the exception of some sparse results obtained for $k=1,2$. We establish the sharp gain in the decay rate for all higher order energies in terms of the first energy, and also obtain the sharp gain of decay rates for the $L^2$ norms of the higher order spatial derivatives. The results concern weighted (in time) and also pointwise (in time) energy decay estimates. We also obtain $L^\infty$ estimates for the solution $u$ in dimension $n=3$. As an application we compute explicit decay rates for all energies which involve the dimension $n$ and the bounds for the coefficients $a(x)$ and $b(x)$ in the case $c (x)=1$ and $h(x,t)=0.$

Keywords:

wave equations with variable coefficients,
hyperbolic diffusion,
linear dissipation,
decay rates,
higher order energy.

Mathematics Subject Classification: Primary: 35L05, 35L15; Secondary: 37L15.

Citation:

Article Metrics

HTML views() PDF downloads(91) Cited by(0)

Higher order energy decay rates for damped wave equations with variable coefficients

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

Higher order energy decay rates for damped wave equations with variable coefficients

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content