Dynamic boundary conditions for time dependent fractional operators on extension domains

Abstract

We consider a parabolic semilinear non-autonomous problem $(\tilde{P})$ for a fractional time dependent operator ${\mathcal{B}}_{\Omega }^{s,t}$ 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.

arXiv comment31 pages; to appear on Advances in Differential Equations