Morawetz Estimates Method for Scattering of Radial Energy Sub-critical Wave Equation

TL;DR

This paper proves that radial solutions to a semi-linear energy sub-critical wave equation in 3D scatter if their initial energy outside a large ball decays sufficiently fast, using an enhanced Morawetz estimate.

Contribution

It introduces a refined Morawetz estimate to establish scattering criteria based on decay rates of initial energy outside a ball.

Findings

Solutions scatter if initial energy decay exceeds a specific rate

Enhanced Morawetz estimate is effective for energy decay analysis

Provides conditions for scattering in radial energy sub-critical wave equations

Abstract

In this short paper we consider a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $p \in (3,5)$. We prove that if the energy of radial initial data $(u_0, u_1)$ outside a ball of radius $r$ centred at the origin decays faster than a certain rate $r^{-\kappa(p)}$, then the corresponding solution $u$ must scatter in both two time directions. The main tool of our proof is a more detailed version of the classic Morawetz estimate.