The perturbed Riemann problem for a balance law

Abstract

We study the asymptotic behaviour of the bounded solutions of a hyperbolic conservation law with source term $$ \partial_t u(x,t)+ \partial_x f(u(x,t)) =g(u(x,t)),\quad x\in\mathbb{R},t\geq 0, $$ where the flux $f$ is convex and the source term $g$ has simple zeros. We assume that the initial value coincides outside a compact set with an initial value of Riemann type. We prove that the solutions converge in general to a sequence of travelling waves delimited by two shock waves. Some of the travelling waves are smooth and connect two consecutive zeros of the source term, while the remaining are discontinuous and oscillate around an unstable zero of the source. However, we prove that for a generic class of initial data the asymptotic profile contains only travelling waves of the first type. We also analyze the rate of convergence of the solutions to the asymptotic profile.