On the Unique Solvability of the Cauchy Problem for One Parabolic Differential-Difference Equation with Spatial Translation

In this paper we study the Cauchy problem for a parabolic differential-difference equation as an operator differential equation in a Banach space. Based on the semigroup theory, we obtain conditions on the parameters in the equation that guarantee the existence of a unique classical solution to the original problem. We also rigorously derive the explicit form of the semigroup as a convolution with a Poisson-type kernel. The reasoning applies to the whole scale of spaces $ H^{s}(𝕉^{n}) $ , which makes it possible to consider quite arbitrary initial data, including those from $ L_{1}(𝕉^{n}) $ .