Creeping fronts in degenerate reaction-diffusion systems

We study systems of reaction-diffusion type for two species in one space dimension and investigate the dynamics in the case where the second species does not diffuse. We consider competing species with two stable equilibria and front solutions that connect the two stable states. A free energy function determines a preferred state. If the diffusive species is preferred, travelling waves may appear. Instead, if the non-diffusive species is preferred, stationary fronts are the only monotone travelling waves. We show that these fronts are unstable and that the non-diffusive species can propagate at a logarithmic rate.