Abstract
This paper is devoted to the study of a semilinear diffusion problem with distributed order fractional derivatives on R-N, which can be used to characterize the ultra-slow diffusion processes with time-dependent logarithmical-law attenuation. We use the resolvents approach to present the local well-posedness of mild solutions belonging to L-r (R-N) (r > 2), in which the L-p - L-q estimates and continuity of the operator are first established. Thcn, under the assumption on the initial value belonging to L-p (R-N), the global well-posedness of mild solutions is derived. Moreover, a decay estimate in L-r norm is included.