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On a wave equation with supercritical interior and boundary sources and damping terms

✍ Scribed by Lorena Bociu; Mohammad Rammaha; Daniel Toundykov

Publisher
John Wiley and Sons
Year
2011
Tongue
English
Weight
348 KB
Volume
284
Category
Article
ISSN
0025-584X

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✦ Synopsis

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well-posed in the finite-energy space H 1 (Ξ©) Γ— L2 (βˆ‚Ξ©) with boundary data in L2 due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non-dissipative and are not locally Lipschitz operators from H 1 (Ξ©) into L2 (Ξ©), or L2 (βˆ‚Ξ©). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.

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