Resumo: We consider a parabolic equation of reaction-diffusion type with a discontinuous nonlinearity, which can be expressed by means of a Heaviside functions as a differential inclusion. We show first that under mild assumptions this equation generates a multivalued semiflow that possesses a global attractor and study its structure, which is an interesting and challenging problem. It is noticeable that our problem is the limit of a sequence of Chafee-Infante problems that undergo an infinite sequence of bifurcations, so it is reasonable to expect that it inherits the structure of the attractor of the Chafee-Infante equation. In fact, we prove that there is an infinite (but countable) number of equilibria and that the sequence of equilibria of the approximative problems converges to the equilibria of the limit problem. Since a Lyapunov function exists, the attractor is characterized by the fixed points and their heteroclinic connections, so a full description of the dynamics is got if we determine which connections exist. We give a partial answer to this question. Finally, if we restrict the semiflow to the positive cone, then nice regularity properties of solutions are obtained. In particular, the structure of the global attractor, in this case, is fully understood.