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 B^s, t_ with Wentzell-type boundary conditions in a possibly non-smooth domain N. We prove existence and uniqueness of the mild solution of the associated semilinear abstract Cauchy problem (P) via an evolution family U (t, ). We then prove that the mild solution of the abstract proble (P) actually solves problem (P) via a generalized fractional Green formula.