Dynamics of second order in time evolution equations with state-dependent delay. (August 2015)

• Record Type: Journal Article
• Authors: Chueshov, Igor; Rezounenko, Alexander
• Is Part Of: Nonlinear analysis. Volume 123/124(2015)
• Page Start: 126
• Page End: 149
• Publication Date: 2015-08
• Subjects: 35R10 -- 35B41 -- 74K20 -- 93C23; Second order evolution equations -- State dependent delay -- Nonlinear plate -- Finite-dimensional attractor; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.04.013
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We deal with a class of second order in time nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in the theory of nonlinear plates. We first prove well-posedness in a certain space of functions which are C 1 in time. In contrast with the first order models with discrete state-dependent delay this result does not require any compatibility conditions. The solutions constructed generate a dynamical system in a C 1 -type space over delay time interval. Our main result shows that this dynamical system possesses compact global and exponential attractors of finite fractal dimension. To obtain this result we adapt the recently developed method of quasi-stability estimates.