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

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.