Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprises, among the others, equations involving both standard and Caputo time-derivative, and diffusion operators as the classic and fractional Laplacian, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators.
We then focus on a parabolic equation with fractional derivatives in time and space that governs the scaling limit of continuous-time random walks with anomalous diffusion.  For these equations, the fundamental solution represents the probability density of finding a particle released at the origin at time 0 at a given position and time.    
Using some estimates of the asymptotic behaviour of the fundamental solution, we evaluate in a heuristic way the probability of the process returning infinite times to the origin.