Energy Distribution of solutions to defocusing semi-linear wave equation in two dimensional space

TL;DR

This paper analyzes the energy distribution of solutions to the defocusing nonlinear wave equation in two dimensions, showing energy disperses at near light speed and establishing decay and scattering results under weaker initial data decay assumptions.

Contribution

It proves that energy in solutions moves to infinity at near light speed, with decay estimates and scattering results under weaker initial decay conditions than previously known.

Findings

Energy moves to infinity at near light speed as time tends to infinity.

Inward/outward energy parts vanish over time, resembling free waves.

Decay estimates and scattering results are established under weaker initial data decay assumptions.

Abstract

We consider finite-energy solutions to the defocusing nonlinear wave equation in two dimensional space. We prove that almost all energy moves to the infinity at almost the light speed as time tends to infinity. In addition, the inward/outward part of energy gradually vanishes as time tends to positive/negative infinity. These behaviours resemble those of free waves. We also prove some decay estimates of the solutions if the initial data decay at a certain rate as the spatial variable tends to infinity. As an application, we prove a couple of scattering results for solutions whose initial data are in a weighted energy space. Our assumption on decay rate of initial data is weaker than previous known scattering results.