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Universal bounds for quasilinear parabolic equations with convection
On the Cauchy problem for differentialdifference parabolic equations with highorder nonlocal terms of general kind
1.  Computing Center, The 4th Clinical Polyclinic of Voronezh City, Russia 394077, Voronezh, Lizyukova 24, Russian Federation 
$\frac{\partial u}{\partial t}= \sum$^{n}_{k,j,m=1}$ a_{kjm}\frac{\partial^2u}{\partial x_k\partial x_j} (x_1,...,x_{m1},x_m+h_{kjm},x_{m+1},...,x_n,t),$
assuming that the operator on the righthand 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 initialvalue function and the investigated solution of the differentialdifference equation tends to zero as $t\to\infty$.
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