Dynamic boundary conditions for time dependent fractional operators on extension domains

We consider a parabolic semilinear non-autonomous problem ( P) for a fractional time dependent operator Bs,t_Ω with Wentzell-type boundary conditions in a possibly non-smooth domain ΩN. We prove existence and uniqueness of the mild solution of the associated semilinear abstract Cauchy problem $(P)$ via an evolution family U(t,τ). We then prove that the mild solution of the abstract proble $(P)$ actually solves problem ( P) via a generalized fractional Green formula.