Asymptotic decay for a one-dimensional nonlinear wave equation

TL;DR

This paper investigates the long-term behavior of solutions to a one-dimensional nonlinear wave equation, showing that solutions exhibit average decay in amplitude over time, contrasting with the non-decaying linear case.

Contribution

It demonstrates that nonlinear solutions in one dimension have average decay in $L^ty$ norm, using Radamacher differentiation theorem, despite linear solutions not decaying.

Findings

Solutions have average $L^ty$ decay as $t o ty$

Linear solutions do not exhibit decay, highlighting nonlinear effects

Radamacher differentiation theorem is used in the analysis

Abstract

We consider the asymptotic behaviour of finite energy solutions to the one-dimensional defocusing nonlinear wave equation $-u_{tt} + u_{xx} = |u|^{p-1} u$, where $p > 1$. Standard energy methods guarantee global existence, but do not directly say much about the behaviour of $u(t)$ as $t \to \infty$. Note that in contrast to higher-dimensional settings, solutions to the linear equation $-u_{tt} + u_{xx} = 0$ do not exhibit decay, thus apparently ruling out perturbative methods for understanding such solutions. Nevertheless, we will show that solutions for the nonlinear equation behave differently from the linear equation, and more specifically that we have the average $L^\infty$ decay $\lim_{T \to +\infty} \frac{1}{T} \int_0^T \|u(t)\|_{L^\infty_x(\R)}\ dt = 0$, in sharp contrast to the linear case. An unusual ingredient in our arguments is the classical Radamacher differentiation theorem that asserts that Lipschitz functions are almost everywhere differentiable.