Almost Sectorial Operators in Fractional Superdiffusion Equations

期刊论文
发表日期：

2024年12月2日

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Eduardo Cuesta,
Rodrigo Ponce

摘要

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