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

TL;DR

This paper studies the behavior of solutions to a one-dimensional nonlinear wave equation as the exponent p approaches infinity, showing convergence to a piecewise limit within [-1,1] under certain initial conditions.

Contribution

It demonstrates the limiting behavior of solutions in the high exponent regime and provides an explicit characterization of the limit function.

Findings

Solutions converge locally uniformly to a piecewise limit in [-1,1]

Limit function can be explicitly computed

Convergence holds under smooth initial data with mild non-degeneracy

Abstract

We investigate the behaviour of solutions $\phi = \phi^{(p)}$ to the one-dimensional nonlinear wave equation $-\phi_{tt} + \phi_{xx} = -|\phi|^{p-1} \phi$ with initial data $\phi(0,x) = \phi_0(x)$, $\phi_t(0,x) = \phi_1(x)$, in the high exponent limit $p \to \infty$ (holding $\phi_0, \phi_1$ fixed). We show that if the initial data $\phi_0, \phi_1$ are smooth with $\phi_0$ taking values in $(-1,1)$ and obey a mild non-degeneracy condition, then $\phi$ converges locally uniformly to a piecewise limit $\phi^{(\infty)}$ taking values in the interval $[-1,1]$, which can in principle be computed explicitly.