#### Vol. 2, No. 2, 2009

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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
##### Abstract

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

##### Keywords
nonlinear wave equation
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/