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 CMFD, 2017, Volume 63, Issue 4, Pages 678–688 (Mi cmfd341)

Asymptotic properties of solutions of two-dimensional differential-difference elliptic problems

A. B. Muravnikab

a JSC Concern "Sozvezdie", 14 Plekhanovskaya, 394018 Voronezh, Russia
b RUDN University, 6 Miklukho-Maklaya st., 117198 Moscow, Russia

Abstract: In the half-plane $\{-\infty<x<+\infty\}\times\{0<y<+\infty\}$, the Dirichlet problem is considered for differential-difference equations of the kind $u_{xx}+\sum_{k=1}^ma_ku_{xx}(x+h_k,y)+u_{yy}=0$, where the amount $m$ of nonlocal terms of the equation is arbitrary and no commensurability conditions are imposed on their coefficients $a_1,…,a_m$ and the parameters $h_1,…,h_m$, determining the translations of the independent variable $x$. The only condition imposed on the coefficients and parameters of the studied equation is the nonpositivity of the real part of the symbol of the operator acting with respect to the variable $x$.
Earlier, it was proved that the specified condition (i. e., the strong ellipticity condition for the corresponding differential-difference operator) guarantees the solvability of the considered problem in the sense of generalized functions (according to the Gel'fand–Shilov definition), a Poisson integral representation of a solution was constructed, and it was proved that the constructed solution is smooth outside the boundary line.
In the present paper, the behavior of the specified solution as $y\to+\infty$ is investigated. We prove the asymptotic closedness between the investigated solution and the classical Dirichlet problem for the differential elliptic equation (with the same boundary-value function as in the original nonlocal problem) determined as follows: all parameters $h_1,…,h_m$ of the original differential-difference elliptic equation are assigned to be equal to zero. As a corollary, we prove that the investigated solutions obey the classical Repnikov–Eidel'man stabilization condition: the solution stabilizes as $y\to+\infty$ if and only if the mean value of the boundary-value function over the interval $(-R,+R)$ has a limit as $R\to+\infty$.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 5-100ÍØ-4479.2014.1 Russian Foundation for Basic Research 17-01-00401

DOI: https://doi.org/10.22363/2413-3639-2017-63-4-678-688

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UDC: 517.929

Citation: A. B. Muravnik, “Asymptotic properties of solutions of two-dimensional differential-difference elliptic problems”, Differential and functional differential equations, CMFD, 63, no. 4, Peoples' Friendship University of Russia, M., 2017, 678–688

Citation in format AMSBIB
\Bibitem{Mur17} \by A.~B.~Muravnik \paper Asymptotic properties of solutions of two-dimensional differential-difference elliptic problems \inbook Differential and functional differential equations \serial CMFD \yr 2017 \vol 63 \issue 4 \pages 678--688 \publ Peoples' Friendship University of Russia \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd341} \crossref{https://doi.org/10.22363/2413-3639-2017-63-4-678-688} 

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This publication is cited in the following articles:
1. A. B. Muravnik, “Elliptic Problems with Nonlocal Potential Arising in Models of Nonlinear Optics”, Math. Notes, 105:5 (2019), 734–746
2. A. B. Muravnik, “Elliptic Differential-Difference Equations in the Half-Space”, Math. Notes, 108:5 (2020), 727–732
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