Porous elastic system with nonlinear damping and sources terms |
| |
Authors: | Mirelson M. Freitas M.L. Santos José A. Langa |
| |
Affiliation: | 1. Institute of Exact and Natural Sciences, Doctoral Program in Mathematics, Federal University of Pará, Augusto corrêa Street, Number 01, 66075-110, Belém PA, Brazil;2. Departamento de Ecuaciones Diferenciales Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain |
| |
Abstract: | We study the long-time behavior of porous-elastic system, focusing on the interplay between nonlinear damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. We also prove the existence of a global attractor. |
| |
Keywords: | 35B40 35B41 37B55 37L30 Porous elastic systems Nonlinear damping and sources Nonlinear semigroups Monotone operators Global attractors |
本文献已被 ScienceDirect 等数据库收录! |
|