Inertial Manifolds for von Karman Plate Equations
One of the contemporary approaches to the study of long-time behavior of infinite-dimensional dynamical systems is based on the concept of inertial manifolds which was introduced in  (see also the monographs [61, 90, 273] and the references therein and also Section 7.6 in Chapter 7). These manifolds are finite-dimensional invariant surfaces that contain global attractors and attract trajectories exponentially fast. Moreover, there is a possibility to reduce the study of limit regimes of the original infinite-dimensional system to solving similar problem for a class of ordinary differential equations. Inertial manifolds are generalizations of center-unstable manifolds and are convenient objects to capture the long-time behavior of dynamical systems. The theory of inertial manifolds is related to the method of integral manifolds (see, e.g., [92, 139, 233]), and has been developed and widely studied for deterministic systems by many authors. All known results concerning existence of inertial manifolds require some gap condition on the spectrum of the linearized problem (see, e.g., [45, 50, 61, 90, 227, 236, 273] and the references therein). Although inertial manifolds have been mainly studied for parabolic-like equations, there are some results for damped second order in time evolution equations arising in nonlinear oscillations theory (see, e.g., [45, 50, 61, 236]). These results rely on the approach originally developed in  for a one-dimensional semilinear wave equation and require the damping coefficient to be large enough. In fact, as indicated in , this requirement is a necessary condition in the case of hyperbolic flows.
KeywordsGlobal Attractor Analytic Semigroup Integral Manifold Plate Equation Inertial Manifold
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