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 , , where is a nonnegative measure on that is periodic in . In the case where is a smooth periodic function, it is known that there exists a travelling wave (more precisely a ``pulsating travelling wave'') with average speed if and only if where is a certain positive number depending on This constant is called the ``minimal speed''. In this paper, we first extend this theory by showing the existence of the minimal speed for any nonnegative measure with period Next we study the question of maximizing under the constraint , where 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

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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:
https://doi.org/10.1090/S0002-9947-2010-04931-1

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.