DECAY ESTIMATES FOR A PERTURBED TWO-TERMS SPACE-TIME FRACTIONAL DIFFUSIVE PROBLEM

In the present paper we consider the Cauchy-type problem associated to the space-time fractional differential equation partial differential tu+ partial differential t beta(-Delta)1-beta u - Delta u = g(t, x), t > 0, x is an element of Rn with beta is an element of (0, 1), where the fractional derivative partial differential t beta is in Caputo sense and (-Delta)1-beta is the fractional Laplace operator of order 1-beta. We provide sufficient conditions on the perturbation g which guarantees that the solution satisfies the same long-time decay estimates of the case g = 0, assuming initial datum in Hs,m for some s > 0 and m is an element of (1, infinity). We apply the obtained results to study the existence of global-in-time solutions to the associated nonlinear problems, partial differential tu+ partial differential t beta(-Delta)1-beta u - Delta u = ( |u|p, backward difference (u|u|p-1), assuming small initial datum in Hs,m and supercritical or critical powers.