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

TL;DR

This paper establishes optimal decay rates for higher order energies of damped wave equations with variable coefficients, extending previous results and providing explicit decay estimates involving the dimension and coefficient bounds.

Contribution

It introduces new estimates that relate higher order energies to the first order energy, addressing an open problem for general order derivatives.

Findings

Optimal decay rates for higher order energies are derived.

Explicit decay estimates depend on dimension and coefficient bounds.

L^ estimates for solutions are obtained.

Abstract

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 $h(x)=0.$