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Asymptotic behavior for a semilinear second order evolution equation


Authors: Chunyou Sun, Lu Yang and Jinqiao Duan
Journal: Trans. Amer. Math. Soc. 363 (2011), 6085-6109
MSC (2010): Primary 35G25, 35B40, 35B41
DOI: https://doi.org/10.1090/S0002-9947-2011-05373-0
Published electronically: May 25, 2011
MathSciNet review: 2817420
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Abstract: This paper is devoted to the qualitative analysis for a second order evolution equation $ u_{tt}-\Delta u-\Delta u_t -\varepsilon \Delta u_{tt}+ f(u)=g(x)$ $ (\varepsilon\in [0,1])$ with critical nonlinearity. Some uniformly (w.r.t. $ \varepsilon\in [0,1]$) asymptotic regularity about the solutions has been established for both $ g(x)\in L^2(\Omega)$ and $ g(x)\in H^{-1}$, which shows that the solutions are exponentially approaching a more regular fixed subset uniformly (w.r.t. $ \varepsilon\in [0,1]$). As an application of this regularity result, a family $ \{\mathcal{E}_{\varepsilon}\}_{\varepsilon\in [0,1]}$ of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with a strongly damped wave equation ( $ \varepsilon =0$), the upper semicontinuity, at $ \varepsilon=0$, of the global attractors has been proved.


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Additional Information

Chunyou Sun
Affiliation: School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China
Email: sunchunyou@gmail.com;sunchy@lzu.edu.cn

Lu Yang
Affiliation: School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China
Email: yanglu@lzu.edu.cn

Jinqiao Duan
Affiliation: Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616
Email: duan@iit.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05373-0
Keywords: Evolution equation of second order, strongly damped wave equation, asymptotic regularity, critical exponent, attractors.
Received by editor(s): September 23, 2009
Received by editor(s) in revised form: March 15, 2010
Published electronically: May 25, 2011
Additional Notes: This work was supported by the NSFC Grants 10601021 and 10926089, the Fund of Physics & Mathematics of Lanzhou University Grants LZULL200801 and LZULL200903, and the Fundamental Research Funds for the Central Universities Grant lzujbky-2009-48
Article copyright: © Copyright 2011 American Mathematical Society

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