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On asymptotics of solutions of parabolic equations with nonlocal high-order terms. (English. Russian original) Zbl 1115.35140
J. Math. Sci., New York 135, No. 1, 2695-2720 (2006); translation from Tr. Semin. Im. I. G. Petrovskogo 25, 143-183 (2005).
Summary: We study the Cauchy problem for second-order differential-difference parabolic equations \[ \frac{\partial u}{\partial t}= \frac{\partial^2u}{\partial x^2}+ \sum_{k=1}^m a_k \frac{\partial^2u}{\partial x^2} (x+h_k,t) \quad\text{in }\mathbb R^1\times(0,\infty) \] containing translation operators acting to the high-order derivatives with respect to spatial variables. We construct the integral representation of the solution and investigate its long-time behavior. We prove theorems on asymptotic closeness of the constructed solution and the Cauchy problem solutions for classical parabolic equations; in particular, conditions of the stabilization of the solution are obtained.

MSC:
35R10 Partial functional-differential equations
35B40 Asymptotic behavior of solutions to PDEs
35C15 Integral representations of solutions to PDEs
35A08 Fundamental solutions to PDEs
35B35 Stability in context of PDEs
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