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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: First, we introduce the concept of pullback -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 -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 -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