Robust attractors for a Kirchhoff-Boussinesq type equation

Zhijian Yang; Na Feng; Yanan Li

doi:10.3934/eect.2020020

Article Contents

2020, Volume 9, Issue 2: 469-486. Doi: 10.3934/eect.2020020

This issue Previous Article Moving and oblique observations of beams and plates Next Article On a Kirchhoff wave model with nonlocal nonlinear damping

Robust attractors for a Kirchhoff-Boussinesq type equation

1.
School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China
2.
College of Science, Zhongyuan University of Technology, No.41, Zhongyuan Road, Zhengzhou 450007, China

^* Corresponding author: Zhijian Yang
^* Corresponding author: Zhijian Yang

Received: October 2018

Revised: January 2019

Early access: December 2019

Published: June 2020

The authors are supported by NNSF of China (Grant No. 11671367)

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Abstract

The paper studies the existence of the pullback attractors and robust pullback exponential attractors for a Kirchhoff-Boussinesq type equation: $ u_{tt}-\Delta u_{t}+\Delta^{2} u = div\Big\{\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\Big\}+\Delta g(u)+f(x,t) $. We show that when the growth exponent $ p $ of the nonlinearity $ g(u) $ is up to the critical range: $ 1\leq p\leq p^*\equiv\frac{N+2}{(N-2)^{+}} $, (ⅰ) the IBVP of the equation is well-posed, and its solution has additionally global regularity when $ t>\tau $; (ⅱ) the related dynamical process $ \{U_f(t,\tau)\} $ has a pullback attractor; (ⅲ) in particular, when $ 1\leq p< p^* $, the process $ \{U_f(t,\tau)\} $ has a family of pullback exponential attractors, which is stable with respect to the perturbation $ f\in \Sigma $ (the sign space).

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Mathematics Subject Classification: Primary: 35B40, 37L15, 37B55; Secondary: 35B41, 35B33, 35B65.

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