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

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  • 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 $h(x,t)=0.$
    Mathematics Subject Classification: Primary: 35L05, 35L15; Secondary: 37L15.

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