Vol. 5, No. 2, 2012

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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

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+1 T0Tu(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.

nonlinear wave equation
Mathematical Subject Classification 2010
Primary: 35L05
Received: 3 November 2010
Revised: 12 January 2011
Accepted: 7 February 2011
Published: 27 August 2012
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