"Front propagation and phase transitions for fractional diffusion equations" - Xavier Cabré (ICREA y Universitat Politècnica de Catalunya)

Resumen:

Long-range or "anomalous" diffusions, such as diffusions given by the fractional powers (-Δ)^α of the Laplacian, attract lately interest in Physics, Biology, and Finance. The fractional powers of the Laplacian are the infinitesimal generators of the symmetric stable Lévy diffusion processes. These - also called Lévy flights - are diffusion processes that combine Brownian motion together with a jump process.

From the mathematical point of view, nonlinear analysis for fractional diffusions has been mostly developed in the last years. In this talk, I will mainly describe recent results concerning front propagation for the nonlinear fractional heat equation, as well as phase transitions for the fractional elliptic Allen-Cahn equation.

In collaboration with J.-M. Roquejore, we study the propagation of fronts for the fractional KPP equation ∂_tu + (-Δ)^αu = u(1-u) in (0,∞) x Rⁿ, 0≤u≤1, with α∈(0,1). By heuristic considerations, some Physics papers predicted that fronts should propagate at exponential speed - in contrast with the classical case α=1 for which there is propagation at a constant KPP speed. In particular, no traveling wave should exist when α<1. In [1] we establish mathematically these results.

In other works, in collaboration with Y. Sire and E. Cinti, we are concerned with the elliptic equation (-Δ)^αu = f(u) in Rⁿ with α∈(0,1), a model being the bistable nonlinearity f(u) = u-u³. The case α=1/2 was studied in [2]. Our main results concern the existence and properties of "layer" (or heteroclinic) solutions of the equation in dimension n=1. We also establish sharp energy estimates for minimizers in Rⁿ.

References:

[1] Cabré, X. and Roquejore, J.-M., Front propagation in Fisher-KPP equations with fractional diffusion, preprint arXiv:0905.1299.

[2] Cabré, X. and Solá-Morales, J., Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58, 2005, 1678-1732.