Well-posedness of delay parabolic equations with unbounded operators acting on delay terms

In the present paper, the well-posedness of the initial value problem for the delay differential equation dv(t)/dt + Av(t) = B(t) v(t -omega) + f (t), t >= 0; v(t) = g(t) (-omega <= t <= 0) in an arbitrary Banach space E with the unbounded linear operators A and B(t) in E with dense domains D(A) subset of D(B(t)) is studied. Two main theorems on well-posedness of this problem in fractional spaces E-alpha are established. In practice, the coercive stability estimates in Holder norms for the solutions of the mixed problems for delay parabolic equations are obtained.