THE HIGH EXPONENT LIMIT p→∞ FOR THE ONE-DIMENSIONAL NONLINEAR WAVE EQUATION

Abstract

We investigate the behaviour of solutions (Formula Presented) to the one-dimensional nonlinear wave equation (Formula Presented) with initial data (Formula Presented) in the high exponent limit p→∞ (holding ϕ0,ϕ1 fixed). We show that if the initial data ϕ0,ϕ1 are smooth with ϕ0 taking values in (-1; 1) and obey a mild nondegeneracy condition, then ϕ converges locally uniformly to a piecewise limit ϕ (∞) taking values in the interval [-1,1], which can in principle be computed explicitly