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

keywords: stabilization of solutions. high-order nonlocal terms Parabolic differential-difference equations integral representation of solutions

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