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

**1.**Matthieu Alfaro, Danielle Hilhorst, and Hiroshi Matano,*The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system*, J. Differential Equations**245**(2008), no. 2, 505–565. MR**2428009**, 10.1016/j.jde.2008.01.014**2.**D. G. Aronson and H. F. Weinberger,*Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation*, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Springer, Berlin, 1975, pp. 5–49. Lecture Notes in Math., Vol. 446. MR**0427837****3.**D. G. Aronson and H. F. Weinberger,*Multidimensional nonlinear diffusion arising in population genetics*, Adv. in Math.**30**(1978), no. 1, 33–76. MR**511740**, 10.1016/0001-8708(78)90130-5**4.**Henri Berestycki and François Hamel,*Front propagation in periodic excitable media*, Comm. Pure Appl. Math.**55**(2002), no. 8, 949–1032. MR**1900178**, 10.1002/cpa.3022**5.**Henri Berestycki, François Hamel, and Lionel Roques,*Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts*, J. Math. Pures Appl. (9)**84**(2005), no. 8, 1101–1146 (English, with English and French summaries). MR**2155900**, 10.1016/j.matpur.2004.10.006**6.**Henri Berestycki, François Hamel, and Nikolai Nadirashvili,*The speed of propagation for KPP type problems. I. Periodic framework*, J. Eur. Math. Soc. (JEMS)**7**(2005), no. 2, 173–213. MR**2127993**, 10.4171/JEMS/26**7.**Henri Berestycki, François Hamel, and Nikolai Nadirashvili,*The speed of propagation for KPP type problems. II. General domains*, J. Amer. Math. Soc.**23**(2010), no. 1, 1–34. MR**2552247**, 10.1090/S0894-0347-09-00633-X**8.**J. Gärtner, M. Freidlin,*On the propagation of concentration waves in periodic and random media*, Soviet Math. Dokl.**20**(1979), 1282-1286.**9.**X. Liang, X. Zhao,*Spreading speeds and travelling waves for abstract monotone semiflows and its application*, preprint.**10.**N. Kinezaki, K. Kawasaki, F. Takasu, N. Shigesada,*Modeling biological invasion into periodically fragmented environments*, Theor. Population Biol.**64**(2003), 291-302.**11.**N. Shigesada, K. Kawasaki,*Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution*, Oxford Univ. Press, Oxford, 1997.**12.**Nanako Shigesada, Kohkichi Kawasaki, and Ei Teramoto,*Traveling periodic waves in heterogeneous environments*, Theoret. Population Biol.**30**(1986), no. 1, 143–160. MR**850456**, 10.1016/0040-5809(86)90029-8**13.**Aizik I. Volpert, Vitaly A. Volpert, and Vladimir A. Volpert,*Traveling wave solutions of parabolic systems*, Translations of Mathematical Monographs, vol. 140, American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by James F. Heyda. MR**1297766****14.**H. F. Weinberger,*Long-time behavior of a class of biological models*, SIAM J. Math. Anal.**13**(1982), no. 3, 353–396. MR**653463**, 10.1137/0513028**15.**Hans F. Weinberger,*On spreading speeds and traveling waves for growth and migration models in a periodic habitat*, J. Math. Biol.**45**(2002), no. 6, 511–548. MR**1943224**, 10.1007/s00285-002-0169-3

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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

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.