Instability of Discontinuous Traveling Waves for Hyperbolic Balance Laws

It is known that scalar hyperbolic conservation laws with source term and periodic initial value have a property of Poincaré–Bendixson type, namely the solutions converge either to a constant state or to a periodic traveling wave, which is necessarily discontinuous. In this paper we show that generically (with respect to theL1topology) the solutions exhibit a behaviour of the former type. We also show that, while the rate of convergence to a constant state is exponential, the convergence to a traveling wave can be arbitrarily slow