On the solvability of parabolic functional differential equations in Banach spaces

In this paper, a parabolic functional differential equation is considered in the spaces $C(0,T;H_p^1(Q))$ for $p$ close to $2$. The transformations of the space argument are supposed to be multiplicators of the Sobolev spaces with a small smoothness exponent. The machinery of the investigation is based on the semigroup theory. In particular, it is proved that the elliptic part of the operator is a generator of a strongly continuous semigroup.