Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay
Result description
We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz in time. Thisallows us to show that the model considered generates an evolution operator semigroup on a certain space of Lipschitz type functions over delay time interval. The operators are closed for all t greater than zero and continuous for t large enough. Our main result shows that the semigroup possesses compact global and exponential attractors of finite fractal dimension. Our argument is based on the recently developed method of quasi-stability estimates and involves some extension of the theory of global attractors for the case of closed evolutions.
Keywords
Parabolic evolution equationsstate-dependent delayglobal attractorfinite-dimensionexponential attractor
The result's identifiers
Result code in IS VaVaI
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay
Original language description
We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz in time. Thisallows us to show that the model considered generates an evolution operator semigroup on a certain space of Lipschitz type functions over delay time interval. The operators are closed for all t greater than zero and continuous for t large enough. Our main result shows that the semigroup possesses compact global and exponential attractors of finite fractal dimension. Our argument is based on the recently developed method of quasi-stability estimates and involves some extension of the theory of global attractors for the case of closed evolutions.
Czech name
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Czech description
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Classification
Type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BC - Theory and management systems
OECD FORD branch
—
Result continuities
Project
GAP103/12/2431: Systems described by partial differential equations with delays
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2015
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Communications on Pure and Applied Analysis
ISSN
1534-0392
e-ISSN
—
Volume of the periodical
14
Issue of the periodical within the volume
5
Country of publishing house
US - UNITED STATES
Number of pages
20
Pages from-to
1685-1704
UT code for WoS article
000365023300005
EID of the result in the Scopus database
2-s2.0-84930637032
Result type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP
BC - Theory and management systems
Year of implementation
2015