Critical asymptotic behaviour for a perturbed conservation law

Abstract

In this paper we study the intermediate asymptotics as t→∞ of nontrivial, nonnegative solutions to the Cauchy problem for the following nonlinear first order equation: u_t+(u^m)_x+u^p=0, with m>1, p=m+1. We prove that for any bounded, nonnegative, compactly supported initial data the asymptotic behaviour is given in first approximation by a universal function W, which is a contracted “N‐wave” with logarithmically decaying mass.