American Institue of Mathematical Sciences

2006, 16(3): 541-561. doi: 10.3934/dcds.2006.16.541

On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind

 1 Computing Center, The 4th Clinical Polyclinic of Voronezh City, Russia 394077, Voronezh, Lizyukova 24, Russian Federation

Received  January 2006 Revised  June 2006 Published  August 2006

We study the Cauchy problem with bounded continuous initial-value functions for the differential-difference equation

$\frac{\partial u}{\partial t}= \sum$nk,j,m=1$a_{kjm}\frac{\partial^2u}{\partial x_k\partial x_j} (x_1,...,x_{m-1},x_m+h_{kjm},x_{m+1},...,x_n,t),$

assuming that the operator on the right-hand side of the equation is strongly elliptic and the coefficients $a_{kjm}$ and $h_{kjm}$ are real. We prove that this Cauchy problem has a unique solution (in the sense of distributions) and this solution is classical in ${\mathbb R}^n \times (0,+\infty),$ find its integral representation, and construct a differential parabolic equation with constant coefficients such that the difference between its classical bounded solution satisfying the same initial-value function and the investigated solution of the differential-difference equation tends to zero as $t\to\infty$.

Citation: Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541
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