Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay

0444705 - ÚTIA 2016 RIV US eng J - Journal Article
Chueshov, I. - Rezunenko, Oleksandr
Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay.
Communications on Pure and Applied Analysis. Roč. 14, č. 5 (2015), s. 1685-1704. ISSN 1534-0392. E-ISSN 1553-5258
R&D Projects: GA ČR GAP103/12/2431
Institutional support: RVO:67985556
Keywords : Parabolic evolution equations * state-dependent delay * global attractor * finite-dimension * exponential attractor
Subject RIV: BC - Control Systems Theory
Impact factor: 0.926, year: 2015
http://library.utia.cas.cz/separaty/2015/AS/rezunenko-0444705.pdf

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. This allows 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.
Permanent Link: http://hdl.handle.net/11104/0247320