Almost Sectorial Operators in Fractional Superdiffusion Equations

Abstract

In this paper the resolvent family {Sα,β(t)}t≥0 ⊂ L(X,Y) generated by an almost sectorial operator A, where α, β > 0, X , Y are complex Banach spaces and its Laplace transform satisfies Sˆ (z) = zα−β(zα − A)−1 is studied. This family of operators α,β allows to write the solution to an abstract initial value problem of time fractional type of order 1 < α < 2 as a variation of constants formula. Estimates of the norm ∥Sα,β(t)∥, as well as the continuity and compactness of Sα,β(t), for t > 0, are shown. Moreover, the Hölder regularity of its solutions is also studied.