The monograph investigates von Kármán evolution equations from the point of view of the existence and uniqueness and asymptotic behavior of a solution. It contains two main parts.
Part I is devoted to well-posedness of various types of dynamic problems for vibrating plates with geometric nonlinearities due to von Kármán’s theory. Before starting with concrete initial-boundary value problems for these types, the authors provide preliminary background needed for the analysis of von Kármán evolution equations. The cases with and without rotational inertia are considered. In the first case the model is recast as a special case of an abstract evolution equation. Nonlinear dissipation both in the interior and on the boundary is treated. In the case without rotational inertia term more subtle tools involving elements of harmonic analysis are used. The coupled structures involving the thermoelastic coupling, acoustic interaction and plates in a potential flow of gas are investigated too.
Part II deals with long-time dynamics. It starts with nice chapters devoted to the foundations of an abstract theory of attractors for autonomous dynamical systems and to long-time behavior of second-order abstract evolution equations; respectively. The results are applied to von Kármán evolution systems with internal and boundary damping. In the first case the existence of attractors and their properties are presented. In the case of boundary damping new approaches combining multiplier techniques are developed. The phenomenon of changing the linearized dynamics from hyperbolic to parabolic appears in the study of the long-time behavior of thermoelastic plates. Three examples of interactive dynamical systems are considered. The first two are acoustic-structure interaction (isothermal and heat generating) and the third one deals with structure-gas-flow interaction. The last chapter deals with existence of inertial manifolds for von Kármán evolution equations.
The authors have published a lot of interesting papers dealing with long-time asymptotics of semilinear wave equations involving vibrating von Kármán plates. The monograph is nicely written and contains results based on new developments in the subject. It can be recommended to experts working in partial differential equations and dynamical systems and also to physicists and engineers interested in the asymptotic analysis of dissipative systems arising in continuum mechanics.