maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2016-3-342-348mais-348Research ArticleОригинальные статьиArticlesСуществование и устойчивость периодических решений уравнения реакция-диффузия в двумерном случаеExistence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional CaseНефедовН. Н.NefedovN. N.

д-р. физ.-мат. наук, профессор, ул. Ленинские горы, д. 1, стр. 2, г. Москва, 119991 Россия

Professor, GSP-1, 1-2 Leninskiye Gory, Moscow, 119991, Russia

nefedov@phys.msu.ruНикулинЕ. И.NikulinE. I.

аспирант, ул. Ленинские горы, д. 1, стр. 2, г. Москва, 119991 Россия

graduate student, GSP-1, 1-2 Leninskiye Gory, Moscow, 119991, Russia

nikulin@physics.msu.ruМосковский государственный университет им. М.В. ЛомоносоваРоссияLomonosov Moscow State UniversityRussian Federation201620062016233342348Copyright © Нефедов Н.Н., Никулин Е.И., 20162016Нефедов Н.Н., Никулин Е.И.Nefedov N.N., Nikulin E.I.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/348

Параболические сингулярно возмущенные задачи активно исследуются в последние годы в связи с большим количеством практических применений: химическая кинетика, синергетика, астрофизика, биология и т.д. В этой работе исследуется сингулярно возмущенная периодическая задача для параболического уравнения реакция-диффузия в двумерном случае. Рассматривается случай существования внутреннего переходного слоя при несбалансированной нелинейности. Внутренний слой локализован вблизи так называемой кривой переходного слоя. Cтроится асимптотическое разложение решения и определяется асимптотика для кривой переходного слоя. Асимптотическое разложение состоит из регулярной части, внутреннего слоя и части пограничного слоя. В этой работе мы сфокусируем внимание на части внутреннего переходного слоя. С целью его описания вводится локальная система координат в окрестности кривой перехода и используются растянутые переменные. Чтобы обосновать таким образом построенную асимптотику, используется асимптотический метод дифференциальных неравенств. Верхнее и нижнее решения строятся путем достаточно сложной модификации асимптотического разложения решения. Асимптотическая устойчивость решения по Ляпунову доказывается с помощью метода сужающихся барьеров. Этот метод базируется на принципе дифференциальных неравенств, и в нем используются верхнее и нижнее решения, которые экспоненциально стремятся к решению задачи. Как результат, решение является локально единственным.Статья публикуется в авторской редакции.

Parabolic singularly perturbed problems have been actively studied in recent years in connection with a large number of practical applications: chemical kinetics, synergetics, astrophysics, biology, and so on. In this work a singularly perturbed periodic problem for a parabolic reaction-diffusion equation is studied in the two-dimensional case. The case when there is an internal transition layer under unbalanced nonlinearity is considered. The internal layer is localised near the so called transitional curve. An asymptotic expansion of the solution is constructed and an asymptotics for the transitional curve is determined. The asymptotical expansion consists of a regular part, an interior layer part and a boundary part. In this work we focus on the interior layer part. In order to describe it in the neighborhood of the transition curve the local coordinate system is introduced and the stretched variables are used. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The upper and lower solutions are constructed by sufficiently complicated modification of the asymptotic expansion of the solution. The Lyapunov asymptotical stability of the solution was proved by using the method of contracting barriers. This method is based on the asymptotic comparison principle and uses the upper and lower solutions which are exponentially tending to the solution to the problem. As a result, the solution is locally unique.The article is published in the authors’ wording.

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The authors declare that there are no conflicts of interest present.