Asymptotic behaviour of solutions to nonlinear shells in a supersonic flow
Asymptotic behaviour of a nonlinear PDE system describing nonlinear vibrations of a shell immersed in a supersonic gas is considered. The model under consideration is a prototype of hyperbolic-like, nondissipative PDE dynamics with nonlinear damping. The main result is the existence of a compact attractor which `attracts' all finite energy (weak) solutions. It is also shown that the Hausdorf dimension of this attractor is finite. The results obtained previously in the literature on related dynamics deal with weakly compact attractors and regular solutions. Thus, a novel contribution of this paper is to obtain compact in the finite energy space attractors for all finite energy solutions (rather than regular).
Numerical Functional Analysis and Optimization
Heyman, W., & Lasiecka, I. (1999). Asymptotic behaviour of solutions to nonlinear shells in a supersonic flow. Numerical Functional Analysis and Optimization, 20 (3), 279-300. https://doi.org/10.1080/01630569908816892