The Perturbed Riemann Problem for a Balance Law

We investigate the asymptotic behavior of the bounded global entropy solutions to the hyperbolic scalar balance law ut+f(u)x=g(u), for a convex flux function f and a source term g with simple zeros. For initial data coinciding outside a compact interval with Riemann data, we describe the generic asymptotic behavior of the solutions. These may converge, as the time goes to infinity, to a sequence of traveling waves bounded by two shock waves. Such traveling waves can be smooth if they connect two consecutive zeros of the source term; otherwise they can be discontinuous and oscillating around an unstable zero. However, we are able to prove that, for generic initial data, the solutions converge to the waves of the first type.