Dynamic boundary conditions for time dependent fractional operators on extension domains

TL;DR

This paper studies a non-autonomous fractional parabolic problem with dynamic boundary conditions on possibly irregular domains, establishing existence, uniqueness, and solution characterization via evolution families and Green formulas.

Contribution

It introduces a framework for fractional operators with boundary conditions that vary over time, extending analysis to non-smooth domains and providing solution existence and uniqueness results.

Findings

Existence and uniqueness of mild solutions for the problem.

Solution characterization via a generalized fractional Green formula.

Application of evolution families to non-autonomous fractional problems.

Abstract

We consider a parabolic semilinear non-autonomous problem $(\tilde P)$ for a fractional time dependent operator $\mathcal{B}^{s,t}_\Omega$ with Wentzell-type boundary conditions in a possibly non-smooth domain $\Omega\subset\mathbb{R}^N$. We prove existence and uniqueness of the mild solution of the associated semilinear abstract Cauchy problem $(P)$ via an evolution family $U(t,\tau)$. We then prove that the mild solution of the abstract problem $(P)$ actually solves problem $(\tilde P)$ via a generalized fractional Green formula.