Unbounded Perturbations of the Generator Domain

Let X, U and Z be Banach spaces such that Z in X (with continuous and dense embedding), L : Z ->X be a closed linear operator and consider closed linear operators G, M : Z -> U. Putting conditions on G and M we show that the operator A = L with domain D(A) ={z∈Z: Gz=Mz} generates a C0- semigroup on X. Moreover, we give a variation of constants formula for the solution of an inhomogeneous problem. Several examples will be given, in particular heat equation with distributed unbounded delay at the boundary condition.