Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains

We investigate quantitative properties of nonnegative solutions to a nonlinear fractional diffusion equation, of degenerate type, posed in a bounded domains, with appropriate homogeneous Dirichlet boundary conditions. The diffusion is driven by a quite general class of linear operators that includes the three most common versions of the fractional Laplacian in a bounded domain with zero Dirichlet boundary conditions; many other examples are included. The nonlinearity is assumed to be increasing and is allowed to be degenerate, the prototype being convex powers. We will present some recent results about existence, uniqueness and a priori estimates for a quite large class of very weak solutions, that we call weak dual solutions.
We will devote special attention to the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our techniques cover also the local case, and provide new results even in this setting.
A surprising instance of this problem is the possible presence of nonmatching powers for the sharp upper and lower boundary behavior. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the elliptic case.
The above results are contained on a series of recent papers in collaboration with A. Figalli, Y. Sire, X. Ros-Oton and J. L. Vazquez.