Almost Sectorial Operators in Fractional Superdiffusion Equations

Abstract

In this paper the resolvent family \(\{S_{\alpha ,\beta }(t)\}_{t\ge 0}\subset \mathcal {L}(X,Y)\) generated by an almost sectorial operator A,  where \(\alpha ,\beta >0,\) X, Y are complex Banach spaces and its Laplace transform satisfies \(\hat{S}_{\alpha ,\beta }(z)=z^{\alpha -\beta }(z^\alpha -A)^{-1}\) is studied. This family of operators allows to write the solution to an abstract initial value problem of time fractional type of order \(1<\alpha <2\) as a variation of constants formula. Estimates of the norm \(\Vert S_{\alpha ,\beta }(t)\Vert ,\) as well as the continuity and compactness of \(S_{\alpha ,\beta }(t)\), for \(t>0\), are shown. Moreover, the Hölder regularity of its solutions is also studied.