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

Abstract

In this paper, we study the asymptotic behavior of the following extensible beam equations: $$ \varepsilon(t) u_{tt}+\Delta^2 u-M\bigg(\int_\Omega |\nabla u|^2\,dx\bigg) \Delta u +\alpha u_t+\varphi (u)=f, \quad t>\tau, $$ 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.