The existence and regularity theory for abstract semilinear time-fractional evolution equations

TL;DR

This paper develops existence, regularity, and differentiability results for abstract time-fractional evolution equations with nonlinear perturbations, extending classical theories to fractional contexts in Banach spaces.

Contribution

It introduces a framework for solving large-time interval problems with Lipschitz nonlinearities and extends blow-up and regularity results to fractional evolution equations.

Findings

Solutions exist for large times independent of Lipschitz constants

Extended blow-up alternative to fractional equations

Proved time differentiability of solutions under smooth perturbations

Abstract

In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We will extend well-known results for standard evolution equations such as the blow-up alternative, to the time-fractional evolution equations. We also prove the differentiability with respect to time of the solution when the perturbation is sufficiently smooth. The differentiability enables us to use the maximum principle. The theory on general Banach spaces enables us to deduce space regularity result easily.