Well-posedness of abstract distributed-order fractional diffusion equations

In this paper, based on distributed-order fractional diffusion equationwe propose the distributed-order fractional abstract Cauchy problem (DFACP)and study the well-posedness of DFACP.Using functional calculus technique, we prove that the general distributed-order fractional operatorgenerates a bounded analytic α-times resolvent operator family or a C₀-semigroup under some suitable conditions.In addition, we reveal the relation between two α-times resolvent familiesgenerated by the sectorial operator A and the special distributed-order fractional operator,p₁Aβ₁+ p₂Aβ₂+… +pₙAβₙ, respectively.