Asymptotic energy distribution of one-dimensional nonlinear wave equation

TL;DR

This paper analyzes the long-term energy distribution of solutions to the one-dimensional nonlinear wave equation, showing energy concentrates near the light cone and providing insights into asymptotic behaviors.

Contribution

It establishes the asymptotic energy distribution near the light cone for the 1D nonlinear wave equation and explores the behavior of solutions with compact initial data.

Findings

Energy concentrates near the light cone as time goes to infinity.

Any light cone eventually contains some energy.

Results apply to both defocusing and focusing cases.

Abstract

In this work we consider the defocusing nonlinear wave equation in one-dimensional space. We show that almost all energy is located near the light cone $|x|=|t|$ as time tends to infinity. We also prove that any light cone will eventually contain some energy. As an application we obtain a result about the asymptotic behaviour of solutions to focusing one-dimensional wave equation with compact-supported initial data.