Instability of discontinuous traveling waves for hyperbolic balance laws (Q678044)

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 the \(L^1\) topology) 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.