Scattering for defocusing energy subcritical nonlinear wave equations

TL;DR

This paper proves that solutions to the defocusing energy subcritical nonlinear wave equation in three dimensions are global and scatter to free waves, provided their critical Sobolev norm remains bounded.

Contribution

It establishes global existence and scattering for energy subcritical nonlinear wave equations under boundedness of the critical Sobolev norm.

Findings

Solutions are global in time for 3 < p < 5.

Solutions scatter to free waves in both time directions.

Bounded critical Sobolev norm ensures scattering.

Abstract

We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the range $3 < p< 5$. We prove that any solution is global-in-time and scatters to free waves in both time directions as long as its critical Sobolev norm stays bounded on the maximal interval of existence.