# The high exponent limit $p o infty$ for the one-dimensional nonlinear wave equation - Mathematics > Analysis of PDEs

The high exponent limit $p o infty$ for the one-dimensional nonlinear wave equation - Mathematics > Analysis of PDEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: We investigate the behaviour of solutions $\phi = \phi^{p}$ to theone-dimensional nonlinear wave equation $-\phi {tt} + \phi {xx} = -|\phi|^{p-1}\phi$ with initial data $\phi0,x = \phi 0x$, $\phi t0,x = \phi 1x$, inthe high exponent limit $p \to \infty$ holding $\phi 0, \phi 1$ fixed. Weshow that if the initial data $\phi 0, \phi 1$ are smooth with $\phi 0$ takingvalues in $-1,1$ and obey a mild non-degeneracy condition, then $\phi$converges locally uniformly to a piecewise limit $\phi^{\infty}$ takingvalues in the interval $-1,1$, which can in principle be computed explicitly.

Author: Terence Tao

Source: https://arxiv.org/