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

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.