Cauchy Problem for a Differential-difference Equation with a Multidimensional Spatial Translation and Summable Initial Function

Abstract: We put to study the Cauchy problem for parabolic differential-difference equations with translations in potentials with respect to spatial independent variables. The initial-value functions are considered to belong to the class of summable functions. The solution of the problem is constructed in a form of a convolution of the kernel of the parabolic differential-difference equation and the initial-value function. The smoothness of the solution and its derivatives is also the subject of the investigation. © 2025 Elsevier B.V., All rights reserved.