Existence and Stability of the Periodic Solution with an Interior Transitional Layer in the Problem with a Weak Linear Advection
In the paper, we study a singularly perturbed periodic in time problem for the parabolic reaction-advection-diffusion equation with a weak linear advection. The case of the reactive term in the form of a cubic nonlinearity is considered. On the basis of already known results, a more general formulat...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2018-02-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/637 |
| _version_ | 1797878009232883712 |
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| author | Nikolay N. Nefedov Egor Ig. Nikulin |
| author_facet | Nikolay N. Nefedov Egor Ig. Nikulin |
| author_sort | Nikolay N. Nefedov |
| collection | DOAJ |
| description | In the paper, we study a singularly perturbed periodic in time problem for the parabolic reaction-advection-diffusion equation with a weak linear advection. The case of the reactive term in the form of a cubic nonlinearity is considered. On the basis of already known results, a more general formulation of the problem is investigated, with weaker sufficient conditions for the existence of a solution with an internal transition layer to be provided than in previous studies. For convenience, the known results are given, which ensure the fulfillment of the existence theorem of the contrast structure. The justification for the existence of a solution with an internal transition layer is based on the use of an asymptotic method of differential inequalities based on the modification of the terms of the constructed asymptotic expansion. Further, sufficient conditions are established to fulfill these requirements, and they have simple and concise formulations in the form of the algebraic equation w(x0,t) = 0 and the condition wx(x0,t) < 0, which is essentially a condition of simplicity of the root x0(t) and ensuring the stability of the solution found. The function w is a function of the known functions appearing in the reactive and advective terms of the original problem. The equation w(x0,t) = 0 is a problem for finding the zero approximation x0(t) to determine the localization region of the inner transition layer. In addition, the asymptotic Lyapunov stability of the found periodic solution is investigated, based on the application of the so-called compressible barrier method. The main result of the paper is formulated as a theorem. |
| first_indexed | 2024-04-10T02:25:58Z |
| format | Article |
| id | doaj.art-4fe8dfea30c8421fa5b5c4c73862398c |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:25:58Z |
| publishDate | 2018-02-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-4fe8dfea30c8421fa5b5c4c73862398c2023-03-13T08:07:29ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172018-02-0125112513210.18255/1818-1015-2018-1-125-132462Existence and Stability of the Periodic Solution with an Interior Transitional Layer in the Problem with a Weak Linear AdvectionNikolay N. Nefedov0Egor Ig. Nikulin1Московский государственный университет им. М.В. ЛомоносоваМосковский государственный университет им. М.В. ЛомоносоваIn the paper, we study a singularly perturbed periodic in time problem for the parabolic reaction-advection-diffusion equation with a weak linear advection. The case of the reactive term in the form of a cubic nonlinearity is considered. On the basis of already known results, a more general formulation of the problem is investigated, with weaker sufficient conditions for the existence of a solution with an internal transition layer to be provided than in previous studies. For convenience, the known results are given, which ensure the fulfillment of the existence theorem of the contrast structure. The justification for the existence of a solution with an internal transition layer is based on the use of an asymptotic method of differential inequalities based on the modification of the terms of the constructed asymptotic expansion. Further, sufficient conditions are established to fulfill these requirements, and they have simple and concise formulations in the form of the algebraic equation w(x0,t) = 0 and the condition wx(x0,t) < 0, which is essentially a condition of simplicity of the root x0(t) and ensuring the stability of the solution found. The function w is a function of the known functions appearing in the reactive and advective terms of the original problem. The equation w(x0,t) = 0 is a problem for finding the zero approximation x0(t) to determine the localization region of the inner transition layer. In addition, the asymptotic Lyapunov stability of the found periodic solution is investigated, based on the application of the so-called compressible barrier method. The main result of the paper is formulated as a theorem.https://www.mais-journal.ru/jour/article/view/637сингулярно возмущенные параболические задачипериодические задачислабая адвекцияуравнения реакция-адвекция-диффузияконтрастные структурывнутренние слоифронтыасимптотические методыдифференциальные неравенстваасимптотическая устойчивость по ляпунову |
| spellingShingle | Nikolay N. Nefedov Egor Ig. Nikulin Existence and Stability of the Periodic Solution with an Interior Transitional Layer in the Problem with a Weak Linear Advection Моделирование и анализ информационных систем сингулярно возмущенные параболические задачи периодические задачи слабая адвекция уравнения реакция-адвекция-диффузия контрастные структуры внутренние слои фронты асимптотические методы дифференциальные неравенства асимптотическая устойчивость по ляпунову |
| title | Existence and Stability of the Periodic Solution with an Interior Transitional Layer in the Problem with a Weak Linear Advection |
| title_full | Existence and Stability of the Periodic Solution with an Interior Transitional Layer in the Problem with a Weak Linear Advection |
| title_fullStr | Existence and Stability of the Periodic Solution with an Interior Transitional Layer in the Problem with a Weak Linear Advection |
| title_full_unstemmed | Existence and Stability of the Periodic Solution with an Interior Transitional Layer in the Problem with a Weak Linear Advection |
| title_short | Existence and Stability of the Periodic Solution with an Interior Transitional Layer in the Problem with a Weak Linear Advection |
| title_sort | existence and stability of the periodic solution with an interior transitional layer in the problem with a weak linear advection |
| topic | сингулярно возмущенные параболические задачи периодические задачи слабая адвекция уравнения реакция-адвекция-диффузия контрастные структуры внутренние слои фронты асимптотические методы дифференциальные неравенства асимптотическая устойчивость по ляпунову |
| url | https://www.mais-journal.ru/jour/article/view/637 |
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