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

G. L. Rossovskii<sup>ab

a</sup> Peoples Friendship University of Russia (RUDN University), Miklukho-Maklaya str. 6, 117198, Moscow, Russia
^b MIREA — Russian Technological University, Vernadsky av. 78, 119454, Moscow, Russia

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.

Keywords: differential–difference operator, parabolic equation, Cauchy problem.

UDC: 517.518

MSC: 35R10, 35K15

Received: 26.11.2024

 English version: Ufa Mathematical Journal, 2025, 17:4, 95–103