Dynamics of a Reaction-diffusion equation with a Discontinuous Nonlinearity

We study the nonlinear dynamics of a reaction-diffusion equation where the nonlinearity presents a discontinuity. We prove the upper semicontinuity of solutions and of the global attractor with respect to smooth approximations of the nonlinear term. We also give a complete description of the set of fixed points and study their stability. Finally, we analyze the existence of heteroclinic connections between the fixed points, obtaining information on the fine structure of the global attractor. AMS Subject Classification: 35B40, 35B41, 35K55, 35K57, 37B25, 58C06