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

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

2008-11-13

arxiv.org/abs/0811.2159v1

Mathematics

Analysis of PDEs

19 pages

Scientific paper

Under appropriate assumptions the energy of wave equations with damping and variable coefficients $c(x)u_{tt}-\hbox{div}(b(x)\nabla u)+a(x)u_t =h(x)$ has been shown to decay. Determining the rate of decay for the higher order energies involving the $k$th order spatial and time derivatives has been an open problem with the exception of some sparse results obtained for $k=1,2,3$. We establish estimates that optimally relate the higher order energies with the first order energy by carefully analyzing the effects of linear damping. The results concern weighted (in time) and also pointwise (in time) energy decay estimates. We also obtain $L^\infty$ estimates for the solution $u$. 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 $h(x)=0.$

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