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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\textit{A. B. Muravnik}, J. Math. Sci., New York 135, No. 1, 2695--2720 (2006; Zbl 1115.35140); translation from Tr. Semin. Im. I. G. Petrovskogo 25, 143--183 (2005)

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