Much recent research is taking place in the area of elliptic and parabolic equations, aimed at understanding the effect of replacing the Laplace operator, and its usual variants, by a fractional Laplacian operator or other similar nonlocal operators, which represent long distance interactions. Linear and nonlinear models are involved.
The lecture will describe some of the progress made by the author and collaborators on the topic of nonlinear fractional heat equations, in particular when the nonlinearities are of porous medium and fast diffusion type. The results cover existence and uniqueness of weak (sometimes classical) solutions, regularity and continuous dependence, positivity, estimates and symmetrization. Special attention is given to the construction of fractional Barenblatt solutions and their asymptotic role.
References:
-L. A. Caffarelli, J. L. Vazquez. “Nonlinear porous medium flow with fractional potential pressure”. Arch. Rat. Mech. Anal., 202 (2011), 537-565.
-Juan Luis Vazquez. “Nonlinear Diffusion with Fractional Laplacian Operators”. in “Nonlinear partial differential equations: the Abel Symposium 2010”, Holden, Helge & Karlsen, Kenneth H. eds., Springer, 2012. Pp. 271-298.
-Juan Luis Vazquez. “Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators”. In “Nonlinear elliptic and parabolic differential equations”, Disc. Cont. Dyn. Syst, to appear in 2014; arXiv:1401.3640.