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We establish new bounds of the Sobolev norms of solutions of semilinear waveequations for data lying in the Hs, s<1, closure of compactly supported datainside a ball of radius R, with R a fixed and positive number. In order to dothat we perform an analysis in the neighborhood of the cone, using an almostShatah-Struwe estimate, an almost conservation law and some estimates forlocalized functions: this allows to prove a decay estimate and establish a lowfrequency estimate of the position of the solution. Then, in order to establisha high frequency estimate of the position and an estimate of the velocity, weuse this decay estimate and another almost conservation law.

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