Attractor for a model of extensible beam with damping on time-dependent space

Fengjuan Meng, Yonghai Wang, Chunxiang Zhao

DOI: http://dx.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


Time-dependent; attractors; extensible beam equations; non-autonomous

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