Research demonstrates the existence of unique solutions for the Cauchy problem in Banach spaces, indicating broader applicability for various initial data.
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 Hˢ(ⁿ) , which makes it possible to consider quite arbitrary initial data, including those from L₁(ⁿ) .
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Rossovskii et al. (2026) studied this question.
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