Analysis & PDE

  • Anal. PDE
  • Volume 5, Number 2 (2012), 411-422.

Asymptotic decay for a one-dimensional nonlinear wave equation

Hans Lindblad and Terence Tao

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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.

Article information

Anal. PDE, Volume 5, Number 2 (2012), 411-422.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L05: Wave equation

nonlinear wave equation


Lindblad, Hans; Tao, Terence. Asymptotic decay for a one-dimensional nonlinear wave equation. Anal. PDE 5 (2012), no. 2, 411--422. doi:10.2140/apde.2012.5.411.

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