Abstract: We establish new bounds of the Sobolev norms of solutions of semilinear wave equations for data lying in the $H^s$, $s<1$, closure of compactly supported data inside a ball of radius $R$, with $R$ a fixed but 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 solution, we use this decay estimate and another almost conservation law.
