Asymptotics and Attractors for Quasilinear Parabolic-Hyperbolic Systems Governing the Motions of Heavily Burdened Deformable Bodies

This paper surveys recent nonlinear analyses of the motion of deformable solids to which are attached heavy rigid bodies. The deformable solids are described by geometrically exact equations of nonlinear viscoelasticity (of strain-rate type), which form quasilinear parabolic-hyperbolic systems. The main emphasis of this paper is on the treatment of problems in which the ratios of the inertias of the deformable body are small with respect to those of the rigid body. Some of these problems possess rigorous asymptotic expansions in a small inertia parameter with the leading terms of the reduced problems governed not by traditional ordinary differential equations for the rigid body, but by equations with memory. Some of these problems admit attractors with the dimensions of the attractors small when the inertia parameters are small. This paper describes a number of open problems.