On the Solvability of Parabolic Functional Differential Equations in Banach Spaces

Abstract

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.