The global attractor for a class of extensible beams with nonlocal weak damping

Chunxiang Zhao; Chunyan Zhao; Chengkui Zhong

doi:10.3934/dcdsb.2019197

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2020, Volume 25, Issue 3: 935-955. Doi: 10.3934/dcdsb.2019197

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The global attractor for a class of extensible beams with nonlocal weak damping

Department of Mathematics, Nanjing University, Nanjing, 210093, China

^* Corresponding author: Chengkui Zhong
^* Corresponding author: Chengkui Zhong

Received: December 31, 2018

Early access: September 2019

Published: March 2020

This work is partly supported by the NSFC (11731005, 11801228) and Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX19-0027)

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Abstract

The goal of this paper is to study the long-time behavior of a class of extensible beams equation with the nonlocal weak damping

$ \begin{eqnarray*} u_{tt}+\Delta^2 u-m(\|\nabla u\|^2)\Delta u +\| u_t\|^{p}u_t+f(u) = h, \rm{in}\; \Omega\times\mathbb{R^{+}}, p\geq0 \end{eqnarray*} $

on a bounded smooth domain $ \Omega\subset\mathbb{R}^{n} $ with hinged (clamped) boundary condition. Under some suitable conditions on the Kirchhoff coefficient $ m(\|\nabla u\|^2) $ and the nonlinear term $ f(u) $, the well-posedness is established by means of the monotone operator theory and the existence of a global attractor is obtained in the subcritical case, where the asymptotic smooothness of the semigroup is verified by the energy reconstruction method.

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Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 37L30.

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