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Higher order energy decay rates for damped wave equations with variable coefficients. (English) Zbl 1181.35024

Summary: Under appropriate assumptions, the energy of wave equations with damping and variable coefficients \(c(x)u_{tt}-\text{div}(b(x)\nabla u)+a(x)u_t =h(x,t)\) 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 \(h(x,t)=0\).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
37L15 Stability problems for infinite-dimensional dissipative dynamical systems
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