Convergence towards discontinuous patterns for a degenerate reaction-diffusion system with a non-monotone term

In this paper, we establish new results on the dynamics of a degenerate reaction-diffusion system with hysteresis. Our model determines the evolution of a biological system characterized by the interaction between a sedentary species and a diffusive species. It can be notably applied in forest ecology and in microbiology. We prove the existence of an infinite family of discontinuous stationary solutions using a generalized Mountain Pass Theorem, and analyze the continuity of this family, through an original method based on the convergence of generalized Clarke gradients. We show that the discontinuity interface of the heterogeneous patterns is very sensitive with respect to a variation of the initial condition. Furthermore, we prove a new theorem of convergence towards discontinuous patterns, which shows that the basin of attraction of each pattern contains a non-trivial set of initial conditions.