Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models

Transactions of the American Mathematical Society

Volumn 362, Issue 10, 2010, Pages 5125-5168

  Shen, Wenxian ^a  

^a AUBURN UNIVERSITY   (United States)

Author keywords

Almost periodic KPP model; Front solution; Generalized propagating speed interval; Principal Floquet bundle; Principal Lyapunov exponent; Spreading speed interval; Traveling wave solution; Variational principle

Indexed keywords

EID: 77956290700     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-10-04950-0     Document Type: Article

References (78)

1. N. D. Alikakos, P. W. Bates, and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805. MR1467460 (99j:35101)
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. Goldstein, Ed.), Lecture Notes in Math., vol.466, Springer-Verlag, New York, 1975, 5-49. MR0427837 (55:867)
3. D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. MR511740 (80a:35013)
4. H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, 101-123,
5. Contemp. Math., 446,
6. Amer. Math. Soc., Providence, RI, 2007. MR2373726 (2009k:35135)
7. H. Berestycki, F. Hamel, and N. Nadirashili, The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213. MR2127993 (2005k:35186)
8. H. Berestycki, F. Hamel, and N. Nadirashili, The speed of propagation for KPP type problems. II. General domains, preprint.
9. H. Berestycki, F. Hamel, and L. Roques, Analysis of periodically fragmented environment model. II. Biological invasions and pulsating traveling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146. MR2155900 (2006d:35123)
10. X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Diff. Eq., 184 (2002), no.2, 549-569. MR1929888 (2003h:35107)
11. X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), no.1, 123-146. MR1981615 (2004b:37175)
12. J. G. Conlon and C. R. Doering, On travelling waves for the stochastic Fisher-Kolmogorov-Petrovsky-Piscunov equation, J. Stat. Phys., 120 (2005), no.3-4, 421-477. MR2182316 (2006j:60064)
13. F. Dumortier, N. Popovic, and T. J. Kaper, The critical wave speed for the Fisher-Kolmogorov-Petrowskii-Piscounov equation with cut-off, Nonlinearity, 20 (2007), no.4, 855-877. MR2307884
14. P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. MR0442480 (56:862)
15. A. M. Fink, "Almost Periodic Differential Equations", Lecture Notes in Mathematics 377, Springer-Verlag, Berlin/Heidelberg/New York, 1974. MR0460799 (57:792)
16. R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7(1937), 335-369.
17. M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and random media, Soviet Math. Dokl., 20 (1979), 1282-1286. MR553200 (81d:80005)
18. A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR0181836 (31:6062)
19. J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), no.3, 489-525. MR2221123 (2007b:34071)
20. F. Hamel, Qualitative properties of monostable pulsating fronts: Exponential decay and monotonicity, prepint.
21. S. Heinze, G. Papanicolaou, and A. Stevens, A variational principle for propagation speeds in inhomogeneous media, SIAM J. Appl. Math., 62 (2001), 129-148. MR1857539 (2002j:35169)
22. D. Henry, "Geometric Theory of Semilinear Parabolic Equations", Lecture Notes in Math. vol.840, Springer-Verlag, Berlin, 1981. MR610244 (83j:35084)
23. J. H. Huang and W. Shen, Speeds of spread and propagation for KPP models in time almost and space periodic media, preprint.
24. W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media. Boundary value problems for functional-differential equations, World Sci. Publ., River Edge, NJ, 1995, 187-199. MR1375475 (97a:35112)
25. W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equation, Comm. Appl. Nonlinear Anal., 1 (1994), no.3, 23-46. MR1295491 (95k:35110)
26. J. Húska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Diff. Eq., 226 (2006), no.2, 541-557. MR2237690 (2007h:35144)
27. J. Húska, Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: The divergence case, Indiana Univ. Math. J., 55 (2006), no.3, 1015-1043. MR2244596 (2007d:35128)
28. J. HŠka and P. Pol?čik, The principal Floquet bundle and exponential separation for linear parabolic equations, J. Dynam. Diff. Eq., 16 (2004), no.2, 347-375. MR2105779 (2006e:35147)
29. J. Húska, P. Poláčik and M. V. Safonov, Harnack inequality, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), no.5, 711-739. MR2348049 (2008k:35211)
30. V. Hutson, W. Shen, and G. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc., 129 (2000), no.6, 1669-1679. MR1814096 (2001m:35243)
31. R. A. Johnson, K. J. Palmer and G. R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), no.1, 1-33. MR871817 (88a:58112)
32. Y. Kametaka, On the nonlinear diffusion equation of Kolmogorov- Petrovskii-Piskunov type, Osaka J. Math., 13 (1976), 11-66. MR0422875 (54:10861)
33. A. Kolmogorov, I. Petrowsky, and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26.
34. L. A. Lewis, B. Li, and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), no.3, 219-233. MR1930975 (2003h:92023)
35. B. Li, H. F. Weinberger, and L.A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), no.1, 82-98. MR2156610 (2006g:92069)
36. X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), no.1, 1-40. MR2270161 (2007i:37144)
37. X. Liang, Y. Yi, and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Diff. Eq., 231 (2006), no.1, 57-77. MR2287877 (2008a:47110)
38. R. Lui, Biological growth and spread modeled by systems of recursions. Math. Biosciences, 93 (1989), 269-312. MR984281 (90g:92069)
39. S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Diff. Eq., 217 (2005), no.1, 54-87. MR2170528 (2006g:35277)
40. A. J. Majda and P. E. Souganidis, Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7 (1994), 1-30. MR1260130 (95e:35180)
41. A. J. Majda and P. E. Souganidis, Flame fronts in a turbulent combustion model with fractal velocity fields, Comm. Pure Appl. Math., 51 (1998), 13371348. MR1639220 (99f:35090)
42. H. Matano, Talks presented at various conferences.
43. J. Mierczyński, and W. Shen, Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations, J. Diff. Eq., 191 (2003), 175-205. MR1973287 (2004h:35232)
44. J. Mierczyński and W. Shen, "Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications," Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 2008.
45. C. Mueller and R. B. Sowers, Random travelling waves for the KPP equation with noise, J. Funct. Anal., 128 (1995), no.2, 439-498. MR1319963 (97a:60083)
46. J. Nolen, M. Rudd, and J. Xin, Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24. MR2142338 (2006h:35129)
47. J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete Cont. Dynam. Syst., 13 (2005), 1217-1234. MR2166666 (2006e:35175)
48. J. Nolen and J. Xin, A variational principle based study of KPP minimal front speeds in random shears, Nonlinearity, 18 (2005), 1655-1675. MR2150348 (2006a:76058)
49. J. Nolen and J. Xin, A variational principle for KPP front speeds in temporally random shear flows, Comm. Math. Phys. 269 (2007), no.2, 493-532. MR2274555 (2007k:35237)
50. J. Nolen and J. Xin, Reaction-diffusion front speeds in spatially-temporally periodic shear flows, Multiscale Model. Simul. 1 (2003), 554-570. MR2029591 (2004j:35154)
51. P. Poláčik, On uniqueness of positive entire solutions and other properties of linear parabolic equations, Discrete Contin. Dyn. Syst., 12 (2005), no.1, 13-26. MR2121246 (2005k:35170)
52. P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynam. Diff. Eq., 5 (1993), no.2, 279-303;
53. Erratum, J. Dynam. Diff. Eq., 6 (1994), no. 1, 245-246. MR1223450 (94d:47064)
54. N. Popovic and T. J. Kaper, Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction-diffusion equations, J. Dynam. Diff. Eq., 18 (2006), no.1, 103-139. MR2220193 (2006m:35196)
55. L. Ryzhik and A Zlatoš, KPP pulsating front speed-up by flows, Commun. Math. Sci., 5 (2007), no.3, 575-593. MR2352332 (2008h:35200)
56. D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355. MR0435602 (55:8561)
57. G. R. Sell, "Topological Dynamics and Ordinary Differential Equations", Van Nostrand Reinhold Company, 1971. MR0442908 (56:1283)
58. W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities, I. Stability and Uniqueness, J. Diff. Eq., 159 (1999), 1-54. MR1726918 (2001k:35157)
59. W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities, II. Existence, J. Diff. Eq., 159 (1999), 55-101. MR1726919 (2001k:35158)
60. W. Shen, Dynamical systems and traveling waves in almost periodic structures. J. Diff. Eq., 169 (2001), 493-548. MR1808473 (2003b:37124)
61. W. Shen, Traveling waves in diffusive random media, J. Dynam Diff. Eq., 16 (2004), 1011-1060. MR2110054 (2005k:35230)
62. W. Shen, Traveling waves in time dependent bistable equations, Diff. and Integral Eq., 19 (2006), 241-278. MR2215558 (2007a:35072)
63. W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 136 (1998), no.647. MR1445493 (99d:34088)
64. W. Shen and Y. Yi, Convergence in almost periodic Fisher and Kolmogorov models, J. Math. Biol. 37 (1998), 84-102. MR1636648 (99k:92037)
65. H. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves foor integral equations and delayed reaction-diffusion models, J. Diff. Eq., 195 (2003), 430-470. MR2016819 (2004h:45007)
66. K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 183 (1978), 453-508. MR509494 (80g:35016)
67. W. A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965), 719-751. MR0187014 (32:4469)
68. H. F. Weinberger, Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396. MR653463 (83f:35019)
69. H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. MR1943224 (2004b:92043a)
70. H. F. Weinberger, M.A. Lewis, and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), no.3, 183-218. MR1930974 (2003h:92029)
71. J. Wu and X. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. Diff. Eq., 135 (1997), no.2, 315-357. MR1441274 (98f:34099)
72. J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delays, J. Dynam. Diff. Eq., 13 (2001), 651-687. MR1845097 (2003a:35114)
73. J. Xin, Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity, J. Dynamics Diff. Eq. 3 (1991) no.4, 541-573. MR1129560 (92k:35150)
74. J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. MR1778352 (2001i:35184)
75. J. Xin, KPP front speeds in random shears and the parabolic Anderson problem, Methods and Applications of Analysis, 10 (2003), 191-198. MR2074747 (2005g:35182)
76. Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Part I. Almost automorphy and almost periodicity, Mem. Amer. Math. Soc. 136 no.647 (1998). MR1445493 (99d:34088)
77. B. Zinner, G. Harris, and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Diff. Eq., 105 (1993), no.1, 46-62. MR1237977 (94k:39034)
78. A. ZlatoŠ, Pulsating front speed-up and quenching of reaction by fast advection, Nonlinearity 20 (2007), 2907-2921. MR2368331 (2008j:35105)