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

TL;DR

This paper develops new bounds for Sobolev norms of solutions to semilinear wave equations with localized initial data, using decay estimates, almost conservation laws, and frequency analysis near the light cone.

Contribution

It introduces novel analytical techniques to control Sobolev norms for solutions with localized data in the subunit Sobolev space, advancing understanding of wave equation behavior.

Findings

Established decay estimates for solutions near the light cone.

Derived low and high frequency bounds for solution position and velocity.

Extended Sobolev norm control to data in the closure of compactly supported functions.

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 fixed and 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 position and an estimate of the velocity, we use this decay estimate and another almost conservation law.