Asymptotic decay for a one-dimensional nonlinear wave equation

Lindblad, Hans; Tao, Terence

doi:10.2140/apde.2012.5.411

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Asymptotic decay for a one-dimensional nonlinear wave equation

Hans Lindblad and Terence Tao

Vol. 5 (2012), No. 2, 411–422

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 $- u_{t t} + u_{x x} = | u |^{p - 1} u$ , 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 $- u_{t t} + u_{x x} = 0$ 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 $lim_{T \to + \infty} \frac{1}{T} \int_{0}^{T} ∥ u (t) ∥_{L_{x}^{\infty} (ℝ)} d t = 0$ , 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.

Keywords

nonlinear wave equation

Mathematical Subject Classification 2010

Primary: 35L05

Milestones

Received: 3 November 2010

Revised: 12 January 2011

Accepted: 7 February 2011

Published: 27 August 2012

Authors

Hans Lindblad
Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 United States

Terence Tao
Department of Mathematics University of California, Los Angeles 405 Hilgard Avenue Los Angeles, CA 90095-1555 United States

Vol. 5, No. 2, 2012