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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity
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by Meihua Yang and Chunyou Sun PDF
Trans. Amer. Math. Soc. 361 (2009), 1069-1101 Request permission

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 op- timal 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
  • ORCID: 0000-0003-3770-7651
  • Email: cysun@amss.ac.cn, sunchunyou@gmail.com
  • 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.
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 1069-1101
  • MSC (2000): Primary 37L05, 35B40, 35B41
  • DOI: https://doi.org/10.1090/S0002-9947-08-04680-1
  • MathSciNet review: 2452835