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$\frac{\partial u}{\partial t}=
\sum$^{n}_{k,j,m=1}$ a_{kjm}\frac{\partial^2u}{\partial x_k\partial x_j}
(x_1,...,x_{m-1},x_m+h_{kjm},x_{m+1},...,x_n,t),$

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
${\mathbb R}^n \times (0,+\infty),$ 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 $t\to\infty$.

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