A variational problem associated with the minimal speed of travelling waves for spatially periodic reactiondiffusion equations
Authors:
Xing Liang, Xiaotao Lin and Hiroshi Matano
Journal:
Trans. Amer. Math. Soc. 362 (2010), 56055633
MSC (2010):
Primary 35K57, 35K55, 35P15; Secondary 92D40, 35B10, 35B30, 35B50, 35K15, 28A25
Published electronically:
June 14, 2010
MathSciNet review:
2661490
Fulltext PDF
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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 AllenCahn equation and the
FitzHughNagumo system, J. Differential Equations 245
(2008), no. 2, 505–565. MR 2428009
(2009f:35144), http://dx.doi.org/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
(55 #867)
 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
(80a:35013), http://dx.doi.org/10.1016/00018708(78)901305
 4.
Henri
Berestycki and François
Hamel, Front propagation in periodic excitable media, Comm.
Pure Appl. Math. 55 (2002), no. 8, 949–1032. MR 1900178
(2003d:35139), http://dx.doi.org/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
(2006d:35123), http://dx.doi.org/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
(2005k:35186), http://dx.doi.org/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
(2010k:35260), http://dx.doi.org/10.1090/S089403470900633X
 8.
J. Gärtner, M. Freidlin, On the propagation of concentration waves in periodic and random media, Soviet Math. Dokl. 20 (1979), 12821286.
 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), 291302.
 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
(87h:92086), http://dx.doi.org/10.1016/00405809(86)900298
 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
(96c:35092)
 14.
H.
F. Weinberger, Longtime behavior of a class of biological
models, SIAM J. Math. Anal. 13 (1982), no. 3,
353–396. MR
653463 (83f:35019), http://dx.doi.org/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
(2004b:92043a), http://dx.doi.org/10.1007/s0028500201693
 1.
 M. Alfaro, D. Hilhorst, H. Matano, The singular limit of the AllenCahn equation and the FitzHughNagumo system, J. Differential Equations 245 (2008), 505565. MR 2428009
 2.
 D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics (J. A. Goldstein, ed.), Lecture Notes in Mathematics, 446, SpringerVerlag, 1975, pp. 549. MR 0427837 (55:867)
 3.
 D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math. 30 (1978), 3376. MR 511740 (80a:35013)
 4.
 H. Berestycki, F. Hamel, Front propagation in periodic excitable media, Communications on Pure and Applied Mathematics, LV (2002), 09491032. MR 1900178 (2003d:35139)
 5.
 H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating traveling fronts, J. Math. Pures Appl. 84 (2005), 11011146. MR 2155900 (2006d:35123)
 6.
 H. Berestycki, F. Hamel, N. Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc. (JEMS) 7 (2005), 173213. MR 2127993 (2005k:35186)
 7.
 H. Berestycki, F. Hamel, N. Nadirashvili, The speed of propagation for KPP type problems. II. General domains, J. Amer. Math. Soc. 23 (2010), 134. MR 2552247
 8.
 J. Gärtner, M. Freidlin, On the propagation of concentration waves in periodic and random media, Soviet Math. Dokl. 20 (1979), 12821286.
 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), 291302.
 11.
 N. Shigesada, K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press, Oxford, 1997.
 12.
 N. Shigesada, K. Kawasaki, E. Teramoto, Traveling periodic waves in heterogeneous environments, Theor. Population Biol. 30 (1986), 143160. MR 850456 (87h:92086)
 13.
 Aizik I. Volpert, Vitaly A. Volpert and Vladimir A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Math. Monographs 140, Amer. Math. Soc. 1994. MR 1297766 (96c:35092)
 14.
 H. F. Weinberger, Longtime behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353396. MR 653463 (83f:35019)
 15.
 H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 (2002), 511548. MR 1943224 (2004b:92043a)
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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.utokyo.ac.jp
DOI:
http://dx.doi.org/10.1090/S000299472010049311
PII:
S 00029947(2010)049311
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.
