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

 

A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations


Authors: Xing Liang, Xiaotao Lin and Hiroshi Matano
Journal: Trans. Amer. Math. Soc. 362 (2010), 5605-5633
MSC (2010): Primary 35K57, 35K55, 35P15; Secondary 92D40, 35B10, 35B30, 35B50, 35K15, 28A25
Published electronically: June 14, 2010
MathSciNet review: 2661490
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Abstract: We consider the equation $ u_t=u_{xx}+b(x)u(1-u)$, $ x\in\mathbb{R}$, where $ b(x)$ is a nonnegative measure on $ \mathbb{R}$ that is periodic in $ x$. In the case where $ b(x)$ is a smooth periodic function, it is known that there exists a travelling wave (more precisely a ``pulsating travelling wave'') with average speed $ c$ if and only if $ c\geq c^*(b),$ where $ c^*(b)$ is a certain positive number depending on $ b.$ This constant $ c^*(b)$ is called the ``minimal speed''. In this paper, we first extend this theory by showing the existence of the minimal speed $ c^*(b)$ for any nonnegative measure $ b$ with period $ L.$ Next we study the question of maximizing $ c^*(b)$ under the constraint $ \int_{[0,L)}b(x)dx=\alpha L$, where $ \alpha$ is an arbitrarily given positive constant. This question is closely related to the problem studied by mathematical ecologists in late 1980s but its answer has not been known. We answer this question by proving that the maximum is attained by periodically arrayed Dirac's delta functions $ \alpha L\sum_{k\in\mathbb{Z}}\delta(x+kL).$


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Additional Information

Xing Liang
Affiliation: Department of Mathematics, University of Science and Technology of China, China
Email: xliang@ustc.edu.cn

Xiaotao Lin
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Japan

Hiroshi Matano
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Japan
Email: matano@ms.u-tokyo.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-04931-1
PII: S 0002-9947(2010)04931-1
Received by editor(s): January 31, 2008
Published electronically: June 14, 2010
Additional Notes: The first author was partially supported by Japan Society of Promotion of Science and NSFC Grant:10871185
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.