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

TL;DR

This paper analyzes a time-fractional non-autonomous semilinear evolution equation $\\partial_t^\alpha u(t)=A(t)u(t)+J(u(t))$ with $\\alpha\in(0,1)$, where $-A(t)$ generates analytic semigroups and satisfies Acquistapace-Terreni conditions. It constructs two evolution families $S_\alpha$ and $P_\alpha$ via a Volterra-based representation, proving ultracontractivity bounds that link $L^p$ to $L^q$ mapping properties and enabling a fixed-point approach to obtain local well-posedness of mild solutions; under suitable smallness conditions on the initial data, these solutions are global in time. The results are specialized to a fractional heat-type problem in non-divergence form on bounded domains, where ultracontractivity is quantified by dimension-dependent estimates, ensuring well-posedness in both subcritical and low-regularity settings. Altogether, the work extends non-autonomous fractional evolution theory by establishing ultracontractivity-driven existence results for semilinear problems and providing a concrete PDE application.

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 $A(t)$.

Theorems & Definitions (17)

• definition 1
• lemma 1: Lemma 6 in hezhouFCAA
• lemma 2: Lemma 8 in hezhouFCAA
• lemma 3: Lemma 10 in hezhouFCAA
• lemma 4: Lemma 11 in hezhouFCAA
• definition 2: see Definition 3 in hezhouFCAA
• theorem 1
• Theorem 3.1
• proof
• definition 3: see Definition 6.1 in hezhouBull