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

摘要

Consider the semilinear wave equations in dimension 3 with a defocusing and superconformal power-type nonlinearity and with data lying in the H~s×H~(s-1) (s<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 H~s norm of the high frequency component of the solution grows like T~(1-s)2+ in a neighborhood of s=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 [3,5].~1.