On control of Sobolev norms for some semilinear wave equations with localized data

Consider the semilinear wave equations in dimension 3 with a defocusing and superconformal power-type nonlinearity and with data lying in the Hs×Hs−1Hs×Hs−1 (s<1s<1) closure of smooth functions that are compactly supported inside a ball with fixed radius. We establish new bounds of the Sobolev norms of the solution. In particular, we prove that the HsHs norm of the high frequency component of the solution grows like T∼(1−s)2+T∼(1−s)2+ in a neighborhood of s=1s=1. In order to do that, we perform an analysis in a neighborhood of the cone, using the finite speed of propagation, an almost Shatah–Struwe estimate [17], an almost conservation law and a low–high frequency decomposition [3] and [5].1