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Pointwise decay for solutions of the 2D Neumann exterior problem for the wave equation. II. (English) Zbl 1072.35110

The long-time asymptotic behaviour of solutions to the Neumann initial boundary value problem for the 2-dimensional homogeneous wave equation in an exterior domain with smooth and convex boundary is considered. The focus lies in controlling the influence of the initial data in terms of their Sobolev norms on the \(L^{\infty}\)-norm of the energy density (energy decay). In an earlier paper of the author a decay of \(O\left(t^{-1/2}\ln^{2}\left(e+t\right)\right)\) is obtained, which is only slightly below the decay rate \(O\left(t^{-1/2}\right)\) for the free space case. The present paper improves on the earlier result in as much as it improves on the dependence on the initial data to make it more suitable for applications to nonlinear problems.
Part I, cf. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. 8, 189–206 (2004), see also Preprint, http://www.dmf.unicatt.it/cgi-bin/preprintserv/semmat/Quad2002n02.

MSC:

35L05 Wave equation
35L20 Initial-boundary value problems for second-order hyperbolic equations
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References:

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