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

Abstract

We investigate the behaviour of solutions $ϕ= ϕ^{(p)}$ to the one-dimensional nonlinear wave equation $-ϕ_{tt} + ϕ_{xx} = -|ϕ|^{p-1} ϕ$ with initial data $ϕ(0,x) = ϕ_0(x)$, $ϕ_t(0,x) = ϕ_1(x)$, in the high exponent limit $p \to \infty$ (holding $ϕ_0, ϕ_1$ fixed). We show that if the initial data $ϕ_0, ϕ_1$ are smooth with $ϕ_0$ taking values in $(-1,1)$ and obey a mild non-degeneracy condition, then $ϕ$ converges locally uniformly to a piecewise limit $ϕ^{(\infty)}$ 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