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

Abstract

We investigate the behaviour of solutions $\varphi ={\varphi }^{(p)}$ to the one-dimensional nonlinear wave equation $-{\varphi }_{tt}+{\varphi }_{xx}=-|\varphi {|}^{p-1}\varphi$ with initial data $\varphi (0,x)={\varphi }_0(x)$, ${\varphi }_t(0,x)={\varphi }_1(x)$, 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 $(-1,1)$ and obey a mild non-degeneracy condition, then $\varphi$ converges locally uniformly to a piecewise limit ${\varphi }^{(\infty )}$ taking values in the interval $[-1,1]$, which can in principle be computed explicitly.

26 pages, 2 figures, submitted, Analysis & PDE. Changes from referee report incorporated