Attractor for a model of extensible beam with damping on time-dependent space
- 8 February 2021
- journal article
- research article
- Published by Uniwersytet Mikolaja Kopernika/Nicolaus Copernicus University in Topological Methods in Nonlinear Analysis
- Vol. 57 (1), 1-29
- https://doi.org/10.12775/tmna.2020.037
Abstract
In this paper, we study the asymptotic behavior of the following extensible beam equations: $$ \varepsilon(t) u_{tt}+\Delta^2 u-M\bigg(\int_\Omega |\nabla u|^2dx\bigg) \Delta u +\alpha u_t+\varphi (u)=f, \quad t> \tau, $$ where $\varepsilon(t)$ is a decreasing function of time vanishing at infinity. After generalizing the abstract results on time dependent space, we establish an invariant time-dependent global attractor for the equation by proving the well-posedness (thereby, the existence of process), dissapativity and the compactness of the process. Our work supplements the theoretical results on time-dependent space and the results on the longtime behavior of the model.
Keywords
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