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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−ω)+f(t), t≥0; v(t)=g(t) (−ω≤t≤0) in an arbitrary Banach space E with the unbounded linear operators A and B(t) in E with dense domains D(A)⊆D(B(t)) is studied. Two main theorems on well-posedness of this problem in fractional spaces Eα are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.

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