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

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731216

Digital Object Identifier
doi:10.2140/apde.2012.5.411

Mathematical Reviews number (MathSciNet)
MR2970713

Zentralblatt MATH identifier
1273.35049

Subjects
Primary: 35L05: Wave equation

Keywords
nonlinear wave equation

Citation

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. https://projecteuclid.org/euclid.apde/1513731216


Export citation

References

  • J. L. Doob, Stochastic processes, Wiley, New York, 1953.
  • D. Lépingle, “La variation d'ordre $p$ des semi-martingales”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36:4 (1976), 295–316.
  • H. Lindblad and A. Soffer, “A remark on asymptotic completeness for the critical nonlinear Klein–Gordon equation”, Lett. Math. Phys. 73:3 (2005), 249–258.
  • M. C. Reed, “Propagation of singularities for non-linear wave equations in one dimension”, Comm. Partial Differential Equations 3:2 (1978), 153–199.
  • T. Tao, Structure and randomness: pages from year one of a mathematical blog, American Mathematical Society, Providence, RI, 2008.
  • T. Tao, “A quantitative version of the Besicovitch projection theorem via multiscale analysis”, Proc. Lond. Math. Soc. $(3)$ 98:3 (2009), 559–584.