Asymptotic profile for a two-terms time fractional diffusion problem

We consider the Cauchy-type problem associated to the time fractional partial differential equation:{partial derivative(t)u + partial derivative(beta)(t) u - Delta u = g(t, x), t > 0, x is an element of R-nu(0,x) = u(0)(x),with beta is an element of (0, 1), where the fractional derivative partial derivative(beta)(t) is in Caputo sense. We provide a sufficient condition on the right-hand term g(t, x) to obtain a solution in C-b ([0, infinity), H-s). We exploit a dissipative-smoothing effect which allows to describe the asymptotic profile of the solution in low space dimension.