Correctness of a problem with initial conditions for parabolic differential-difference equations with shifts of time argument

In this paper we consider the problem with initial conditions on semiaxis for differential-difference equation of parabolic type with deviation of time argument. We obtain sufficient condition of correct solvability of the problem in Sobolev space with exponential weight. In terms of spectrum of operator of the problem, we obtain necessary conditions of correct solvability, sufficient conditions of absence of solution and sufficient conditions of solutions nonuniqueness.