The high exponent limit p →∞ for the one-dimensional nonlinear wave equation

Tao, Terence

doi:10.2140/apde.2009.2.235

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The high exponent limit $p \to \infty$ for the one-dimensional nonlinear wave equation

Terence Tao

Vol. 2 (2009), No. 2, 235–259

DOI: 10.2140/apde.2009.2.235

Abstract

We investigate the behaviour of solutions $ϕ = ϕ^{(p)}$ to the one-dimensional nonlinear wave equation $- ϕ_{t t} + ϕ_{x x} = - | ϕ |^{p - 1} ϕ$ with initial data $ϕ (0, x) = ϕ_{0} (x)$ , $ϕ_{t} (0, x) = ϕ_{1} (x)$ , in the high exponent limit $p \to \infty$ (holding $ϕ_{0}, ϕ_{1}$ fixed). We show that if the initial data $ϕ_{0}, ϕ_{1}$ are smooth with $ϕ_{0}$ taking values in $(- 1, 1)$ and obey a mild nondegeneracy condition, then $ϕ$ converges locally uniformly to a piecewise limit $ϕ^{(\infty)}$ taking values in the interval $[- 1, 1]$ , which can in principle be computed explicitly.

Keywords

nonlinear wave equation

Mathematical Subject Classification 2000

Primary: 35L15

Milestones

Received: 22 January 2009

Revised: 13 February 2009

Accepted: 5 April 2009

Published: 1 May 2009

Authors

Terence Tao
Department of Mathematics University of California, Los Angeles Mathematics Department Los Angeles, CA 90095-1555 United States
http://ftp.math.ucla.edu/~tao/

Vol. 2, No. 2, 2009