The evolution fractional p-Laplacian equation in [formula omitted]. Fundamental solution and asymptotic behaviour

Abstract

We consider the natural time-dependent fractional p-Laplacian equation posed in the whole Euclidean space, with parameters p>2 and s∈(0,1) (fractional exponent). We show that the Cauchy Problem for data in the Lebesgue Lq spaces is well posed, and show that the solutions form a family of non-expansive semigroups with regularity and other interesting properties. As main results, we construct the self-similar fundamental solution for every mass value M, and prove that general finite-mass solutions converge towards that fundamental solution having the same mass, and convergence holds in all Lq spaces. A number of additional properties and estimates complete the picture.