We report on recent progress in the study of nonlinear diffusion equations involving
nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous …

We establish the mean field convergence for systems of points evolving along the gradient
flow of their interaction energy when the interaction is the Coulomb potential or a super …

The fractional Laplacian (-Δ)^α/2 is a nonlocal operator which depends on the parameter α
and recovers the usual Laplacian as α→2. A numerical method for the fractional Laplacian is …

We describe the mathematical theory of diffusion and heat transport with a view to including
some of the main directions of recent research. The linear heat equation is the basic …

The analysis of nonlocal discrete equations driven by fractional powers of the discrete
Laplacian on a mesh of size h> 0 (− Δ h) su= f, for u, f: Z h→ R, 0< s< 1, is performed. The …

RESEARCH PAPER FRACTIONAL CALCULUS FOR POWER FUNCTIONS AND EIGENVALUES
OF THE FRACTIONAL LAPLACIAN Bart lomiej Dyda Abstract We Page 1 RESEARCH PAPER …

A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as
a porous medium equation whose pressure law is nonlinear and nonlocal. We show the …

We establish the existence, uniqueness and main properties of the fundamental solutions for
the fractional porous medium equation introduced in [51]. They are self-similar functions of …

We significantly expand the number of functions whose image under the fractional Laplace
operator can be computed explicitly. In particular, we show that the fractional Laplace …

We consider a model of fractional diffusion involving a natural nonlocal version of the p-
Laplacian operator. We study the Dirichlet problem posed in a bounded domain Ω of RN …