Radiation fields and non-radiative solutions to the energy sub-critical wave equations

TL;DR

This paper develops a comprehensive theory of radiation fields for free waves in energy sub-critical wave equations and demonstrates that non-radiative solutions asymptotically behave like free waves, without energy assumptions.

Contribution

It introduces a new framework for radiation fields in the energy sub-critical regime and links non-radiative solutions to free wave asymptotics in critical Sobolev spaces.

Findings

Radiation fields theory for free waves in sub-critical case

Non-radiative solutions asymptotically resemble free waves

Theory applies without energy assumptions

Abstract

Radiation field and channel of energy method have become important tools in the study of nonlinear wave equations in recent years. In this work we give basic theory of radiation fields of free waves in the energy sub-critical case. We also show that the asymptotic behaviours of non-radiative solutions to a wide range of non-linear wave equations resemble those of non-radiative free waves. Our theory is completely given in the critical Sobolev spaces of the corresponding nonlinear wave equation and avoids any assumption on the energy of the solutions.