We study non-autonomous semilinear evolution equations

$\partial_t^\alpha u=A(t)u(t)+J(u(t))$ $t\in (0,T)$

$u(0)=u_0$

with fractional-in-time derivatives, governed by sectorial operators A(t) that satisfy the classical Acquistapace–Terreni conditions.

These conditions ensure the well-posedness of the associated linear evolution families despite the lack of time invariance. Our analysis introduces the fractional solution operators S_alpha(t, τ) and P_alpha(t, τ), for which we establish ultracontractivity estimates that generalize classical heat-kernel bounds to the fractional and non-autonomous setting. These estimates provide a crucial tool for controlling nonlinearities.

Building on this linear foundation, we address the semilinear equation through fixed-point arguments formulated in weighted function spaces adapted to fractional temporal behavior. We prove local well-posedness for a broad class of nonlinearities, requiring only localized Lipschitz continuity and suitable growth conditions. Furthermore, under additional smallness assumptions on the initial data, we obtain global-in-time existence results. These findings extend and refine existing theories for both autonomous fractional equations and classical parabolic problems.

To illustrate the applicability of our abstract theory, we discuss a fractional heat equation with time-dependent, uniformly elliptic operators in non-divergence form. This example highlights how the developed framework accommodates PDEs with variable coefficients, nonlinear effects, and fractional temporal dynamics.

Reference: S. Creo and M. R. Lancia, Non-autonomous semilinear fractional evolution equations: well-posedness and ultracontractivity results, 2025.