Absence and Blow-Up of Solutions to Singular Perturbation Problems in the Case of Exchange of Stabilities
We consider some singular perturbation problems in the case where a degenerate equation has intersecting roots (this case is also referred to as ‘ the exchange of stabilities’). Such problems often occur as models in chemical kinetics. There are lots of works that establish the existence and asympto...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2016-10-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/392 |
| _version_ | 1811154011721564160 |
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| author | M. A. Terentyev |
| author_facet | M. A. Terentyev |
| author_sort | M. A. Terentyev |
| collection | DOAJ |
| description | We consider some singular perturbation problems in the case where a degenerate equation has intersecting roots (this case is also referred to as ‘ the exchange of stabilities’). Such problems often occur as models in chemical kinetics. There are lots of works that establish the existence and asymptotic behavior of solutions to such problems. Due to exchange of stabilities, a typical solution approaches the non-smooth (but continuous) composite root of the degenerate equation as the perturbation parameter gets smaller. In a number of problems a regular part of the perturbative term dominates the singular one, so an additional condition on the regular part is needed to improve the stability of a composite root in the vicinity of the intersection point. Inversion of that condition results in a loss or a blow-up of the solution for sufficiently small values of the perturbation parameter. We prove some results of this kind by means of the nonlinear capacity argument and discuss their role in developing numerical algorithms for the problems under consideration. |
| first_indexed | 2024-04-10T02:24:00Z |
| format | Article |
| id | doaj.art-43ff56ec2bd741bb96eb95ee1e669597 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:24:00Z |
| publishDate | 2016-10-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-43ff56ec2bd741bb96eb95ee1e6695972023-03-13T08:07:34ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-10-0123558759410.18255/1818-1015-2016-5-587-594328Absence and Blow-Up of Solutions to Singular Perturbation Problems in the Case of Exchange of StabilitiesM. A. Terentyev0Московский государственный университет имени М.В. Ломоносова, 119991, Россия, ГСП–1, г. Москва, Ленинские горы, д. 1, стр. 2We consider some singular perturbation problems in the case where a degenerate equation has intersecting roots (this case is also referred to as ‘ the exchange of stabilities’). Such problems often occur as models in chemical kinetics. There are lots of works that establish the existence and asymptotic behavior of solutions to such problems. Due to exchange of stabilities, a typical solution approaches the non-smooth (but continuous) composite root of the degenerate equation as the perturbation parameter gets smaller. In a number of problems a regular part of the perturbative term dominates the singular one, so an additional condition on the regular part is needed to improve the stability of a composite root in the vicinity of the intersection point. Inversion of that condition results in a loss or a blow-up of the solution for sufficiently small values of the perturbation parameter. We prove some results of this kind by means of the nonlinear capacity argument and discuss their role in developing numerical algorithms for the problems under consideration.https://www.mais-journal.ru/jour/article/view/392малый параметрсингулярные возмущениянеизолированный кореньсмена устойчивостинесуществованиеразрушениенелинейная ёмкость |
| spellingShingle | M. A. Terentyev Absence and Blow-Up of Solutions to Singular Perturbation Problems in the Case of Exchange of Stabilities Моделирование и анализ информационных систем малый параметр сингулярные возмущения неизолированный корень смена устойчивости несуществование разрушение нелинейная ёмкость |
| title | Absence and Blow-Up of Solutions to Singular Perturbation Problems in the Case of Exchange of Stabilities |
| title_full | Absence and Blow-Up of Solutions to Singular Perturbation Problems in the Case of Exchange of Stabilities |
| title_fullStr | Absence and Blow-Up of Solutions to Singular Perturbation Problems in the Case of Exchange of Stabilities |
| title_full_unstemmed | Absence and Blow-Up of Solutions to Singular Perturbation Problems in the Case of Exchange of Stabilities |
| title_short | Absence and Blow-Up of Solutions to Singular Perturbation Problems in the Case of Exchange of Stabilities |
| title_sort | absence and blow up of solutions to singular perturbation problems in the case of exchange of stabilities |
| topic | малый параметр сингулярные возмущения неизолированный корень смена устойчивости несуществование разрушение нелинейная ёмкость |
| url | https://www.mais-journal.ru/jour/article/view/392 |
| work_keys_str_mv | AT materentyev absenceandblowupofsolutionstosingularperturbationproblemsinthecaseofexchangeofstabilities |