5/7/2010
Xavier Cabré
ICREA and Universitat Politècnica de Catalunya
Front propagation and phase transitions for fractional diffusion equations
Long-range or \anomalous" di ffusions, such as di ffusions given by the frac- tional powers (-\Delta)^s of the Laplacian, attract lately interest in Physics, Biol- ogy, and Finance. From the mathematical point of view, nonlinear analysis for fractional di ffusions is being developed actively in the last years. In this talk, I will describe recent results concerning front propagation for the nonlinear fractional KPP heat equation, \partial_t tu+(-\Delta)^su = u(1-u) in (0;1)Rn, 0 u 1, with s 2 (0; 1). In collaboration with J.-M. Roquejo re, we establish that fronts propagate at exponential speed |in contrast with the classical case s = 1 for which there is propagation at a constant KPP speed. I will also describe works in collaboration with Y. Sire and E. Cinti. They concern the fractional elliptic Allen-Cahn equation (-\Delta)^su = f(u) in Rn with s 2 (0; 1), a model being the bistable nonlinearity f(u) = u - u3. Our main results concern the existence and properties of \layer" or heteroclinic solutions, as well as of minimizers of the equation.