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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity


Authors: Meihua Yang and Chunyou Sun
Journal: Trans. Amer. Math. Soc. 361 (2009), 1069-1101
MSC (2000): Primary 37L05, 35B40, 35B41
Published electronically: September 29, 2008
MathSciNet review: 2452835
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Abstract: This paper is dedicated to analyzing the dynamical behavior of strongly damped wave equations with critical nonlinearity in locally uniform spaces. After proving the global well-posedness, we first establish the asymptotic regularity of the solutions which appears to be optimal and the existence of a bounded (in $ H^2_{lu}(\mathbb{R}^N)\times H^1_{lu}(\mathbb{R}^N)$) subset which attracts exponentially every initial $ H^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N)$-bounded set with respect to the $ H^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N)$-norm. Then, we show there is a $ (H ^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N), H^1_\rho(\mathbb{R}^N)\times H^1_\rho(\mathbb{R}^N))$-global attractor, which reflects the strongly damped property of $ \Delta u_t$ to some extent.


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Additional Information

Meihua Yang
Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China – and – Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China
Email: yangmeih@gmail.com

Chunyou Sun
Affiliation: Department of Mathematics, Lanzhou University, Lanzhou, 730000, People’s Republic of China – and – Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China
Email: cysun@amss.ac.cn, sunchunyou@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04680-1
PII: S 0002-9947(08)04680-1
Keywords: Strongly damped wave equation, locally uniform spaces, critical exponent, asymptotic regularity, attractors.
Received by editor(s): May 18, 2007
Published electronically: September 29, 2008
Additional Notes: This work was supported by the NSFC Grants 10601021 and 10726024 and the China Postdoctoral Science Foundation.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.