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Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator


Authors: Shuang Liu, Yuan Lou, Rui Peng and Maolin Zhou
Journal: Proc. Amer. Math. Soc. 147 (2019), 5291-5302
MSC (2010): Primary 35P15; Secondary 35K87, 35B10
DOI: https://doi.org/10.1090/proc/14653
Published electronically: July 1, 2019
MathSciNet review: 4021088
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Abstract: We investigate the effect of frequency on the principal eigenvalue of a time-periodic parabolic operator with Dirichlet, Robin, or Neumann boundary conditions. The monotonicity and asymptotic behaviors of the principal eigenvalue with respect to the frequency parameter are established. Our results prove a conjecture raised by Hutson, Michaikow, and Poláčik.


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Additional Information

Shuang Liu
Affiliation: Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, People’s Republic of China
Email: liushuangnqkg@ruc.edu.cn

Yuan Lou
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Email: lou@math.ohio-state.edu

Rui Peng
Affiliation: School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, People’s Republic of China
Email: pengrui_seu@163.com

Maolin Zhou
Affiliation: Department of Mathematics, School of Science and Technology, University of New England, Armidale, New South Wales 2341, Australia
Email: zhouutokyo@gmail.com

DOI: https://doi.org/10.1090/proc/14653
Keywords: Time-periodic parabolic operator, principal eigenvalue, frequency, monotonicity, asymptotics.
Received by editor(s): January 13, 2019
Received by editor(s) in revised form: March 26, 2019
Published electronically: July 1, 2019
Additional Notes: The first author was partially supported by the NSFC grant No. 11571364 and the Outstanding Innovative Talents Cultivation Funded Programs 2018 of Renmin University of China.
The second author was partially supported by the NSF grant DMS-1411176.
The third author was partially supported by the National Science Foundation (NSF) of China (grants no. 11671175 and 11571200), the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (grant no. PPZY2015A013), and the Qing Lan Project of Jiangsu Province.
The fourth author was partially supported by the Australian Research Council (grant no. DE170101410).
Communicated by: Wenxian Shen
Article copyright: © Copyright 2019 American Mathematical Society