In [11, 16, 17], the first boundary-value problem for parabolic differential-difference equations with translations with respect to spatial variables was considered. In [12], it is found that problems of the above kind are related to nonlocal problems. The second boundary-value problem for parabolic differential-difference equations was originally considered in [10]. Regions of dimension n > 1 are considered in those papers. In [9], the first boundary-value problem was studied for n = 1. It is proved in the above papers that, unlike parabolic equations, the smoothness of solutions may be broken inside the region even in the case where the initial function is infinitely differentiable.
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