Behavior of Solutions to the Cauchy Problem for a Parabolic Differential-Difference Equation with Orthogonal Shifts: Comparison of the L₁(Rⁿ) and L₂(Rⁿ) Initial Data Cases

In this paper we study the Cauchy problem for a second-order parabolic differential-difference equation in the case of a superposition of multidimensional orthogonal spatial shifts. A classical solution to the original equation is constructed in a form of a convolution of a Poisson-type kernel with the initial function. Two qualitatively different cases of the functional class of the initial function are considered: summable and square-summable. The resulting solutions, along with their derivatives, are examined and compared in terms of their qualitative characteristics. Estimates are obtained that describe the behavior of the solutions at large time values. We also obtain explicit sufficient conditions that relate the coefficients of the equation and guarantee that the solution decays at infinity.