**Tristan Roy (Kyoto University)**

Title: On control of sobolev norms with localized data

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 positive number.
In order to do that we perform an analysis in a 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.