On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind

We study the Cauchy problem with bounded continuous initial-value functions for the differential-difference equation [GRAPHICS] assuming that the operator on the right-hand side of the equation is strongly elliptic and the coefficients a(kjm) and h(kjm) are real. We prove that this Cauchy problem has a unique solution (in the sense of distributions) and this solution is classical in R-n x (0, + infinity), find its integral representation, and construct a differential parabolic equation with constant coefficients such that the difference between its classical bounded solution satisfying the same initial-value function and the investigated solution of the differential-difference equation tends to zero as t -> infinity.