On Control Of Sobolev Norms For Some Semilinear Wave Equations With Localized Data

Tristan Roy

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arXiv · 1204.3038

On Control Of Sobolev Norms For Some Semilinear Wave Equations With Localized Data

Tristan Roy

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Abstract

We establish new bounds of the Sobolev norms of solutions of semilinear wave equations for data lying in the Hs, s<1, closure of compactly supported data inside a ball of radius R, with R a fixed and positive number. In order to do that we perform an analysis in the neighborhood of the cone, using an almost Shatah-Struwe estimate, an almost conservation law and some estimates for localized functions: this allows to prove a decay estimate and establish a low frequency estimate of the position of the solution. Then, in order to establish a high frequency estimate of the position and an estimate of the velocity, we use this decay estimate and another almost conservation law.

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Tristan Roy. 2012-04-13. On Control Of Sobolev Norms For Some Semilinear Wave Equations With Localized Data. https://arxiv.org/abs/1204.3038

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