On the Energy Subcritical, Non-linear Wave Equation with Radial Data for $p\in (3,5)$

TL;DR

This paper proves that radial solutions to the 3D energy-subcritical nonlinear wave equation with p in (3,5) are globally well-posed and scatter, using compactness, decay estimates, and energy channel methods.

Contribution

It establishes global existence and scattering for radial solutions in the energy-subcritical regime, extending understanding of wave equations with nonlinearities in this range.

Findings

Radial solutions are globally defined in time.

Radial solutions scatter in the energy space.

The methods combine compactness, decay estimates, and energy channels.

Abstract

In this paper, we consider the wave equation in 3-dimensional space with an energy-subcritical nonlinearity, either in the focusing or defocusing case. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined in time and scatters. The proof depends on the compactness/rigidity argument, decay estimates for radial, "compact" solutions, gain of regularity arguments and the "channel of energy" method.