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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ H^{s}(𝕉^{n}) $</EquationSource> </InlineEquation>, which makes it possible to consider quite arbitrary initial data, including those from&#xa0;<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ L_{1}(𝕉^{n}) $</EquationSource> </InlineEquation>.</p>

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

  • 期刊论文

  • 发表日期：

    2026年3月28日

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  • G. L. Rossovskii,
  • L. E. Rossovskii

摘要

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}) $ .