On diffusion equations driven by nonlinear and nonlocal operators

We  report  on the theory of evolution equations that combine a strongly nonlinear parabolic character with the presence of fractional operators representing long-range interaction effects, mainly of fractional Laplacian type. Examples include nonlocal porous media equations and fractional p-Laplacian operators appearing in a number of variants. 

Recent work concerns the time-dependent fractional p-Laplacian equation with parameter p>1 and fractional exponent 0<s<1. It is the gradient flow corresponding to the Gagliardo–Slobodeckii fractional energy. Our main interest is the asymptotic behavior of solutions posed in the whole Euclidean space, which is given by a kind of Barenblatt solution whose existence relies on a delicate analysis. The superlinear and sublinear ranges involve different analysis and results.