Cauchy problem for parabolic equations with multiple spatial translations and summable initial functions

Abstract

We consider the Cauchy problem for parabolic differential-difference equations with multiple spatial translations in lower order terms. The function in the initial condition is supposed to be summable. The solution to the problem is constructed as the convolution of the kernel of parabolic equation with the initial function. We study the behavior and smoothness of the solution and its derivatives for large time.