Well-posed initial problem for parabolic differential-difference equations with shifts of time argument

In the present paper we study an initial problem on half-axis for differential-difference equations of parabolic type with deviations of time argument. We establish conditions under which the problem is well-posed in the Sobolev spaces with exponential weight. In terms of spectrum of operator of the problem, we obtain necessary conditions for its well-posing, and sufficient conditions for absence of its solutions and for their non-uniqueness. © 2015, Allerton Press, Inc.