Ricerc@Sapienza

In this paper, we consider the porous medium equation and establish a relationship between its Kompanets-Zel'dovich-Barenblatt solution u(xd,t),xd∈Rd,t>0 and random flights. The time-rescaled version of u(xd, t) is the fundamental solution of the Euler-Poisson-Darboux equation, which governs the distribution of random flights performed by a particle whose displacements have a Dirichlet probability distribution and choosing directions uniformly on a d-dimensional sphere.

We study the metrics of constant Q-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation −Deltaw = e^(nw) − c δ_0 on R^n, under a finite volume condition. We analyze the asymptotic behavior at infinity and the existence of solutions for every n ≥ 3 also in a supercritical regime. Finally, we state some open problems.

We study the asymptotic behavior of anomalous fractional diffusion processes in bad domains via
the convergence of the associated energy forms.We introduce the associated Robin–Venttsel’ problems for
the regional fractional Laplacian. We provide a suitable notion of fractional normal derivative on irregular
sets via a fractional Green formula as well as existence and uniqueness results for the solution of the
Robin–Venttsel’ problem by a semigroup approach. Submarkovianity and ultracontractivity properties of
the associated semigroup are proved.