Non-autonomous semilinear fractional evolution equations: well-posedness and ultracontractivity results

TL;DR

This paper studies time-fractional semilinear parabolic equations with time-dependent operators, establishing local and global existence of solutions through ultracontractivity estimates, advancing understanding of fractional evolution equations.

Contribution

It introduces new well-posedness results for fractional evolution equations with time-dependent operators under Acquistapace-Terreni conditions.

Findings

Proved local existence of mild solutions.

Established global existence under certain conditions.

Derived ultracontractivity estimates for fractional evolution families.

Abstract

We consider a time-fractional semilinear parabolic abstract Cauchy problem for a time-dependent sectorial operator $A(t)$ which satisfies the Acquistapace-Terreni conditions. We first prove local existence results for the mild solution of the problem at hand. Then we prove, under suitable assumptions on the initial datum, that the solution is also global in time. This is achieved by proving ultracontractivity estimates for the fractional evolution families associated with the operator $A(t)$.