Asymptotic behavior for a semilinear second order evolution equation
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- by Chunyou Sun, Lu Yang and Jinqiao Duan PDF
- Trans. Amer. Math. Soc. 363 (2011), 6085-6109 Request permission
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.References
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Additional Information
- Chunyou Sun
- Affiliation: School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China
- ORCID: 0000-0003-3770-7651
- 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
- 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
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2817420