New trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems |
| |
Authors: | Fatiha Alabau-Boussouira |
| |
Institution: | INRIA Equipe-projet CORIDA et L.M.A.M., CNRS-UMR 7122, Université de Metz, Ile du Saulcy 57045 Metz Cedex 01, France |
| |
Abstract: | We present an approach based on comparison principles for energy and interpolation properties to derive lower energy estimates for nonlinearly either locally damped or boundary damped vibrating systems. We show how the dissipation relation provides strong information on the asymptotic behavior of the energy of solutions. The geometrical situations are either one-dimensional, or radial two-dimensional or three-dimensional for annulus domains. We also consider the case of general domains, but in this case, for solutions with bounded velocities in time and space. In all these cases, the nonlinear damping function is assumed to have arbitrary (strictly sublinear) growth at the origin. We give results for strong solutions and stronger lower estimates for smoother solutions. The results are presented in two forms, either on the side of energy comparison principles, or through time-pointwise lower estimates. Under additional geometric assumptions, we give the resulting lower and upper estimates for four representative examples of damping functions. We further give a “weak” lower estimate (in the sense of a certain lim supt→∞) and an upper estimate of the velocity for smoother solutions in case of general damping functions and for radial, as well as multi-dimensional domains. We also discuss these estimates in the framework of optimality, which is not proved here, and indicate open problems raised by these results. |
| |
Keywords: | 34G10 35B35 35B37 35L90 93D15 93D20 |
本文献已被 ScienceDirect 等数据库收录! |
|