A Stability Technique for Evolution Partial Differential Equations: A Dynamical Systems Approach
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Boston, MA
Birkhäuser Boston
2004
|
| Series: | Progress in Nonlinear Differential Equations and Their Applications
56 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | common feature is that these evolution problems can be formulated as asymptotically small perturbations of certain dynamical systems with better-known behaviour. Now, it usually happens that the perturbation is small in a very weak sense, hence the difficulty (or impossibility) of applying more classical techniques. Though the method originated with the analysis of critical behaviour for evolution PDEs, in its abstract formulation it deals with a nonautonomous abstract differential equation (NDE) (1) Ut = A(u) + C(u, t), t > 0, where u has values in a Banach space, like an LP space, A is an autonomous (time-independent) operator and C is an asymptotically small perturbation, so that C(u(t), t) ~ ° as t ~ 00 along orbits {u(t)} of the evolution in a sense to be made precise, which in practice can be quite weak. We work in a situation in which the autonomous (limit) differential equation (ADE) Ut = A(u) (2) has a well-known asymptotic behaviour, and we want to prove that for large times the orbits of the original evolution problem converge to a certain class of limits of the autonomous equation. More precisely, we want to prove that the orbits of (NDE) are attracted by a certain limit set [2* of (ADE), which may consist of equilibria of the autonomous equation, or it can be a more complicated object |
| Physical Description: | 1 Online-Ressource (XIX, 377p. 10 illus) |
| ISBN: | 9781461220503 9781461273967 |
| DOI: | 10.1007/978-1-4612-2050-3 |
Staff View
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| 245 | 1 | 0 | |a A Stability Technique for Evolution Partial Differential Equations |b A Dynamical Systems Approach |c by Victor A. Galaktionov, Juan Luis Vázquez |
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| 500 | |a common feature is that these evolution problems can be formulated as asymptotically small perturbations of certain dynamical systems with better-known behaviour. Now, it usually happens that the perturbation is small in a very weak sense, hence the difficulty (or impossibility) of applying more classical techniques. Though the method originated with the analysis of critical behaviour for evolution PDEs, in its abstract formulation it deals with a nonautonomous abstract differential equation (NDE) (1) Ut = A(u) + C(u, t), t > 0, where u has values in a Banach space, like an LP space, A is an autonomous (time-independent) operator and C is an asymptotically small perturbation, so that C(u(t), t) ~ ° as t ~ 00 along orbits {u(t)} of the evolution in a sense to be made precise, which in practice can be quite weak. We work in a situation in which the autonomous (limit) differential equation (ADE) Ut = A(u) (2) has a well-known asymptotic behaviour, and we want to prove that for large times the orbits of the original evolution problem converge to a certain class of limits of the autonomous equation. More precisely, we want to prove that the orbits of (NDE) are attracted by a certain limit set [2* of (ADE), which may consist of equilibria of the autonomous equation, or it can be a more complicated object | ||
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Record in the Search Index
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| author_facet | Galaktionov, Victor A. |
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| author_sort | Galaktionov, Victor A. |
| author_variant | v a g va vag |
| building | Verbundindex |
| bvnumber | BV042419971 |
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| collection | ZDB-2-SMA ZDB-2-BAE |
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| dewey-full | 515.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
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| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4612-2050-3 |
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| id | DE-604.BV042419971 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:05Z |
| institution | BVB |
| isbn | 9781461220503 9781461273967 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027855388 |
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| physical | 1 Online-Ressource (XIX, 377p. 10 illus) |
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| publishDate | 2004 |
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| publisher | Birkhäuser Boston |
| record_format | marc |
| series | Progress in Nonlinear Differential Equations and Their Applications |
| series2 | Progress in Nonlinear Differential Equations and Their Applications |
| spelling | Galaktionov, Victor A. Verfasser aut A Stability Technique for Evolution Partial Differential Equations A Dynamical Systems Approach by Victor A. Galaktionov, Juan Luis Vázquez Boston, MA Birkhäuser Boston 2004 1 Online-Ressource (XIX, 377p. 10 illus) txt rdacontent c rdamedia cr rdacarrier Progress in Nonlinear Differential Equations and Their Applications 56 common feature is that these evolution problems can be formulated as asymptotically small perturbations of certain dynamical systems with better-known behaviour. Now, it usually happens that the perturbation is small in a very weak sense, hence the difficulty (or impossibility) of applying more classical techniques. Though the method originated with the analysis of critical behaviour for evolution PDEs, in its abstract formulation it deals with a nonautonomous abstract differential equation (NDE) (1) Ut = A(u) + C(u, t), t > 0, where u has values in a Banach space, like an LP space, A is an autonomous (time-independent) operator and C is an asymptotically small perturbation, so that C(u(t), t) ~ ° as t ~ 00 along orbits {u(t)} of the evolution in a sense to be made precise, which in practice can be quite weak. We work in a situation in which the autonomous (limit) differential equation (ADE) Ut = A(u) (2) has a well-known asymptotic behaviour, and we want to prove that for large times the orbits of the original evolution problem converge to a certain class of limits of the autonomous equation. More precisely, we want to prove that the orbits of (NDE) are attracted by a certain limit set [2* of (ADE), which may consist of equilibria of the autonomous equation, or it can be a more complicated object Mathematics Global analysis (Mathematics) Differential equations, partial Materials Hydraulic engineering Partial Differential Equations Analysis Continuum Mechanics and Mechanics of Materials Engineering Fluid Dynamics Mathematik Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Evolutionsgleichung (DE-588)4129061-6 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Evolutionsgleichung (DE-588)4129061-6 s Dynamisches System (DE-588)4013396-5 s 1\p DE-604 Vázquez, Juan Luis Sonstige oth Progress in Nonlinear Differential Equations and Their Applications 56 (DE-604)BV036582883 56 https://doi.org/10.1007/978-1-4612-2050-3 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Galaktionov, Victor A. A Stability Technique for Evolution Partial Differential Equations A Dynamical Systems Approach Progress in Nonlinear Differential Equations and Their Applications Mathematics Global analysis (Mathematics) Differential equations, partial Materials Hydraulic engineering Partial Differential Equations Analysis Continuum Mechanics and Mechanics of Materials Engineering Fluid Dynamics Mathematik Partielle Differentialgleichung (DE-588)4044779-0 gnd Evolutionsgleichung (DE-588)4129061-6 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4129061-6 (DE-588)4013396-5 |
| title | A Stability Technique for Evolution Partial Differential Equations A Dynamical Systems Approach |
| title_auth | A Stability Technique for Evolution Partial Differential Equations A Dynamical Systems Approach |
| title_exact_search | A Stability Technique for Evolution Partial Differential Equations A Dynamical Systems Approach |
| title_full | A Stability Technique for Evolution Partial Differential Equations A Dynamical Systems Approach by Victor A. Galaktionov, Juan Luis Vázquez |
| title_fullStr | A Stability Technique for Evolution Partial Differential Equations A Dynamical Systems Approach by Victor A. Galaktionov, Juan Luis Vázquez |
| title_full_unstemmed | A Stability Technique for Evolution Partial Differential Equations A Dynamical Systems Approach by Victor A. Galaktionov, Juan Luis Vázquez |
| title_short | A Stability Technique for Evolution Partial Differential Equations |
| title_sort | a stability technique for evolution partial differential equations a dynamical systems approach |
| title_sub | A Dynamical Systems Approach |
| topic | Mathematics Global analysis (Mathematics) Differential equations, partial Materials Hydraulic engineering Partial Differential Equations Analysis Continuum Mechanics and Mechanics of Materials Engineering Fluid Dynamics Mathematik Partielle Differentialgleichung (DE-588)4044779-0 gnd Evolutionsgleichung (DE-588)4129061-6 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Mathematics Global analysis (Mathematics) Differential equations, partial Materials Hydraulic engineering Partial Differential Equations Analysis Continuum Mechanics and Mechanics of Materials Engineering Fluid Dynamics Mathematik Partielle Differentialgleichung Evolutionsgleichung Dynamisches System |
| url | https://doi.org/10.1007/978-1-4612-2050-3 |
| volume_link | (DE-604)BV036582883 |
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