Finite-dimensional global attractors  for parabolicnonlinear equations with state-dependent delay

Igor Chueshov; Alexander V. Rezounenko

doi:10.3934/cpaa.2015.14.1685

Article Contents

2015, Volume 14, Issue 5: 1685-1704. Doi: 10.3934/cpaa.2015.14.1685

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

Igor Chueshov^1, and
Alexander V. Rezounenko^2,

1.
Department of Mechanics and Mathematics, Karazin Kharkov National University, Kharkov, 61022
2.
Department of Mechanics and Mathematics, V.N.Karazin Kharkiv National University, 4, Svobody Sqr., Kharkiv, 61077

Received: September 30, 2014

Revised: January 31, 2015

Published: August 2015

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Abstract

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 $S_t$ on a certain space of Lipschitz type functions over delay time interval. The operators $S_t$ are closed for all $t\ge 0$ and continuous for $t$ large enough. Our main result shows that the semigroup $S_t$ 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.

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Mathematics Subject Classification: Primary: 35R10, 35B41; Secondary: 93C23.

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