Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Kernel sections of multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains


Authors: Yejuan Wang and Shengfan Zhou
Journal: Quart. Appl. Math. 67 (2009), 343-378
MSC (2000): Primary 34B40, 35K57
DOI: https://doi.org/10.1090/S0033-569X-09-01150-0
Published electronically: March 25, 2009
MathSciNet review: 2514639
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Abstract | References | Similar Articles | Additional Information

Abstract: First, we introduce the concept of pullback $ \omega$-limit compactness for multi-valued processes, as an extension of the similar concept in the autonomous and nonautonomous framework. Next, we present the necessary and sufficient conditions (pullback dissipativeness and pullback $ \omega$-limit compactness) for the existence of a nonempty local bounded kernel (kernel sections are all compact, invariant and pullback attracting) of an infinite dimensional multi-valued process. In addition, we prove a result ensuring the existence of a uniform attractor and the uniform forward attraction of the inflated kernel sections of a family of multi-valued processes under the general assumptions of point dissipativeness and uniform $ \omega$-limit compactness. Finally, we illustrate the abstract theory with a nonlinear reaction-diffusion model in an unbounded domain.


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Additional Information

Yejuan Wang
Affiliation: School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China
Email: wangyj@lzu.edu.cn, wangyjmath@yahoo.com.cn

Shengfan Zhou
Affiliation: Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, People’s Republic of China
Email: zhoushengfan@yahoo.com

DOI: https://doi.org/10.1090/S0033-569X-09-01150-0
Keywords: Multi-valued process, kernel section, nonautonomous reaction-diffusion equations in unbounded domains
Received by editor(s): February 13, 2008
Published electronically: March 25, 2009
Additional Notes: Y. J. Wang was supported by the National Natural Science Foundation of China under Grant 10801066 and the Fundamental Research Fund for Physics and Mathematics of Lanzhou University (LZULL200802).
S. Zhou was supported by the National Natural Science Foundation of China under Grant 10771139, the Innovation Program of Shanghai Municipal Education Commission under Grant 08ZZ70, and the Foundation of Shanghai Normal University under Grant DYL200803.
Article copyright: © Copyright 2009 Brown University

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