Appl Math Optim 46:179–206 (2002)
2002 Springer-Verlag New York Inc.
Inertial Manifolds for von Karman Plate Equations
and Irena Lasiecka
Department of Mathematics and Mechanics, Kharkov University,
Kharkov 310077, Ukraine
Department of Mathematics, University of Virginia,
Charlottesville, VA 22903, USA
Abstract. Inertial manifolds associated with nonlinear plate models governed by
dynamical von Karman equations are considered. Three different dissipative mech-
anisms are discussed: viscous, structural and thermal damping. Though the systems
considered are subject to some dissipation, the overall dynamics may not be dissi-
pative. This means that the energy may not be decreasing.
The main result of the paper establishes the existence of an inertial manifold
subject to the spectral gap condition for linearized problems. The validity of the
spectral gap condition depends on the geometry of the domain and the type of
damping. It is shown that the spectral gap condition holds for plates of rectangular
shape. In the case of viscous damping, which is associated with hyperbolic-like
dynamics, it is also required that the damping parameter be sufﬁciently large. This
last requirement is not needed for other types of dissipation considered in the paper.
Key Words. von Karman equations, Long-time behaviour, Inertial manifolds,
Thermal damping, Viscous damping.
AMS Classiﬁcation. 40A20, 74K20, 74H40, 35Q72.
One of the modern approaches to the study of the long-time behaviour of inﬁnite-
dimensional dynamical systems is based on the concept of an inertial manifold (IM) (see,
e.g. , ,  and the references therein). These manifolds are ﬁnite-dimensional
Igor Chueshov was partially supported by INTAS Grant 2000/899 and USNAS Grant No. 9911017.
Irena Lasiecka was partially supported by National Science Foundation Grant DMS-0104305, Army Research
Ofﬁce Grant DAAD19-02-1-0179 and USNAS Grant No. 9911017.