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

Key Points

• The aim is to establish unique solvability conditions for a parabolic differential-difference equation.
• Utilized operator differential equations within a Banach space.
• Applied semigroup theory to derive conditions for unique solutions.
• Derived an explicit semigroup form as a convolution with a Poisson-type kernel.
• Considered a scale of spaces to accommodate arbitrary initial data.
• Conditions were identified that ensure the existence of a unique classical solution.
• The explicit form of the semigroup was rigorously derived.
• The reasoning is applicable to various initial data, including those from L1(ℝ^n).