Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data. (English) Zbl 1421.35203

Summary: This paper is concerned with weighted energy estimates for solutions to wave equation \(\partial_t^2u-\Delta u+a(x)\partial_tu=0\) with space-dependent damping term \(a(x)=|x|^{-\alpha}\) \((\alpha\in [0,1])\) in an exterior domain \(\Omega\) having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polynomials are given and these decay rates are almost sharp, even when the initial data do not have compact support in \(\Omega\). The crucial idea is to use special solution of \(\partial_tu=|x|^\alpha\Delta u\) including Kummer’s confluent hypergeometric functions.


35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
47B25 Linear symmetric and selfadjoint operators (unbounded)
35B45 A priori estimates in context of PDEs
Full Text: DOI arXiv


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