José Valero

Universidad Miguel Hernández de Elche, Spain

Title: A Hartman-Grobman theorem for a reaction-diffusion equation with discontinuous nonlinearity

Abstract: We study a reaction-diffusion equation governed by a Heaviside-type function. This problem appears as the limit of a sequence of Chafee-Infante problems that have undergo all the typical bifurcation cascade of these type of problems. Hence, the fixed points are of the same kind with the difference that there is an infinite (but countable) number of them. Using a suitable linearized equation around the non-zero fixed points involving the delta Dirac function we establish the stability properties of the equilibria. Moreover, we prove that there are non-zero eigenvalues in the linearized equations and that the eigenspaces generated by the positive and negative eigenvalues are tangent to the unstable and stable sets of the fixed point. Therefore, we prove that the non-zero fixed points satisfy the saddle-point property.