Strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation

Yue Sun; Zhijian Yang; Yanxia Qu

doi:10.3934/dcds.2022116

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2022, Volume 42, Issue 12: 5613-5636. Doi: 10.3934/dcds.2022116

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Strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation

1.
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

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

Received: August 2021

Revised: July 2022

Early access: August 2022

Published: December 2022

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Abstract

In this paper, we investigate the global well-posedness and the existence of strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation in $ \Omega\subset{\mathbb R}^N $:

$ u_{tt}+\Delta ^2u+\Delta ^2u_t+\Delta \phi (\Delta u) = g(x), $

with the hinged boundary condition. We show that (i) when the nonlinearity $ \phi $ is quasi-monotone and is of at most the critical growth: $ 1\leq p\leq p^{*}: = \frac{N+2}{(N-2)^{+}} \ (N\geq 2) $ and $ g = 0 $, the model has in phase space $ {\mathcal H} = V_3\times L^2 $ a trivial global and exponential attractor, respectively. (ii) In particular when $ N = 1 $, without any polynomial growth restriction for $ \phi $, the model has a strong global and a strong exponential attractor, respectively. These results deepen and extend the related researches on this topic in recent literature [16,22]. The method developed here allows us to establish the existence of the strong global and exponential attractor for this nonlinear model.

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Mathematics Subject Classification: Primary: 37L30, 35B33; Secondary: 35B40, 35B41, 35B65.

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