Stability of parabolic equations with unbounded operators acting on delay terms

In this article, we study the stability of the initial value problem for the delay differential equation $$\displaylines{ \frac{dv(t)}{dt}+Av(t)=B(t)v(t-\omega )+f(t),\quad t\geq 0,\cr v(t)=g(t)\quad (-\omega \leq t\leq 0) }$$ in a Banach space E with the unbounded linear operators A and B(t) with dense domains $D(A)\subseteq D(B(t))$. We establish stability estimates for the solution of this problem in fractional spaces $E_{\alpha }$. Also we obtain stability estimates in Holder norms for the solutions of the mixed problems for delay parabolic equations with Neumann condition with respect to space variables.