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2012 Asymptotic decay for a one-dimensional nonlinear wave equation
Hans Lindblad, Terence Tao
Anal. PDE 5(2): 411-422 (2012). DOI: 10.2140/apde.2012.5.411

Abstract

We consider the asymptotic behaviour of finite energy solutions to the one-dimensional defocusing nonlinear wave equation utt+uxx=|u|p1u, where p>1. Standard energy methods guarantee global existence, but do not directly say much about the behaviour of u(t) as t. Note that in contrast to higher-dimensional settings, solutions to the linear equation utt+uxx=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 decay limT+1T0Tu(t)Lx()dt=0, in sharp contrast to the linear case. An unusual ingredient in our arguments is the classical Rademacher differentiation theorem that asserts that Lipschitz functions are almost everywhere differentiable.

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Hans Lindblad. Terence Tao. "Asymptotic decay for a one-dimensional nonlinear wave equation." Anal. PDE 5 (2) 411 - 422, 2012. https://doi.org/10.2140/apde.2012.5.411

Information

Received: 3 November 2010; Revised: 12 January 2011; Accepted: 7 February 2011; Published: 2012
First available in Project Euclid: 20 December 2017

zbMATH: 1273.35049
MathSciNet: MR2970713
Digital Object Identifier: 10.2140/apde.2012.5.411

Subjects:
Primary: 35L05

Keywords: Nonlinear wave equation

Rights: Copyright © 2012 Mathematical Sciences Publishers

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