Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional Case
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 equ...
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Yaroslavl State University
2016-06-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/348 |
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| author | N. N. Nefedov E. I. Nikulin |
| author_facet | N. N. Nefedov E. I. Nikulin |
| author_sort | N. N. Nefedov |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-1ff9ea90ac754002b5d1afdf556b4b4c |
| institution | Kabale University |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2016-06-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-1ff9ea90ac754002b5d1afdf556b4b4c2025-08-20T03:44:17ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-06-0123334234810.18255/1818-1015-2016-3-342-348305Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional CaseN. N. Nefedov0E. I. Nikulin1Lomonosov Moscow State UniversityLomonosov Moscow State UniversityParabolic 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.https://www.mais-journal.ru/jour/article/view/348reaction-diffusionsingular perturbationssmall parameterinterior layersunbalanced reactionboundary layersdifferential inequalitiesupper and lower solutions |
| spellingShingle | N. N. Nefedov E. I. Nikulin Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional Case Моделирование и анализ информационных систем reaction-diffusion singular perturbations small parameter interior layers unbalanced reaction boundary layers differential inequalities upper and lower solutions |
| title | Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional Case |
| title_full | Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional Case |
| title_fullStr | Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional Case |
| title_full_unstemmed | Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional Case |
| title_short | Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional Case |
| title_sort | existence and stability of periodic solutions for reaction diffusion equations in the two dimensional case |
| topic | reaction-diffusion singular perturbations small parameter interior layers unbalanced reaction boundary layers differential inequalities upper and lower solutions |
| url | https://www.mais-journal.ru/jour/article/view/348 |
| work_keys_str_mv | AT nnnefedov existenceandstabilityofperiodicsolutionsforreactiondiffusionequationsinthetwodimensionalcase AT einikulin existenceandstabilityofperiodicsolutionsforreactiondiffusionequationsinthetwodimensionalcase |