How hysteresis produces discontinuous patterns in degenerate reaction-diffusion systems

In this paper, we study the asymptotic behaviour of the solutions to a degenerate reaction-diffusion system. This system admits a continuum of discontinuous stationary solutions due to the effect of a hysteresis process, but only one discontinuous stationary solution is compatible with a principle of preservation of locally invariant regions. Using a macroscopic mass effect which guarantees that fast particles help slow particles to displace, we establish a novel result of convergence of a non trivial set of trajectories towards a discontinuous pattern.