Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion

Lim, Tau Shean; Zlatoš, Andrej

doi:10.1090/tran/6602

Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion
HTML articles powered by AMS MathViewer

by Tau Shean Lim and Andrej Zlatoš PDF

Trans. Amer. Math. Soc. 368 (2016), 8615-8631 Request permission

Abstract:

We prove existence of and construct transition fronts for a class of reaction-diffusion equations with spatially inhomogeneous Fisher-KPP type reactions and non-local diffusion. Our approach is based on finding these solutions as perturbations of appropriate solutions to the linearization of the PDE at zero. Our work extends a method introduced by one of us to study such questions in the case of classical diffusion.

References

Fuensanta Andreu-Vaillo, José M. Mazón, Julio D. Rossi, and J. Julián Toledo-Melero, Nonlocal diffusion problems, Mathematical Surveys and Monographs, vol. 165, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. MR 2722295, DOI 10.1090/surv/165
Peter W. Bates, Paul C. Fife, Xiaofeng Ren, and Xuefeng Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal. 138 (1997), no. 2, 105–136. MR 1463804, DOI 10.1007/s002050050037
H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, H. Berestycki and Y. Pomeau, eds., Kluwer, Dordrecht, 2003.
Henri Berestycki and François Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55 (2002), no. 8, 949–1032. MR 1900178, DOI 10.1002/cpa.3022
Henri Berestycki and François Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math. 65 (2012), no. 5, 592–648. MR 2898886, DOI 10.1002/cpa.21389
Henri Berestycki, Grégoire Nadin, Benoit Perthame, and Lenya Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity 22 (2009), no. 12, 2813–2844. MR 2557449, DOI 10.1088/0951-7715/22/12/002
Jack Carr and Adam Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2433–2439. MR 2052422, DOI 10.1090/S0002-9939-04-07432-5
Jérôme Coville, Harnack type inequality for positive solution of some integral equation, Ann. Mat. Pura Appl. (4) 191 (2012), no. 3, 503–528. MR 2958346, DOI 10.1007/s10231-011-0193-2
Jérôme Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl. (4) 185 (2006), no. 3, 461–485. MR 2231034, DOI 10.1007/s10231-005-0163-7
Jérôme Coville, Juan Dávila, and Salomé Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013), no. 2, 179–223. MR 3035974, DOI 10.1016/j.anihpc.2012.07.005
Jerome Coville and Louis Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 4, 727–755. MR 2345778, DOI 10.1017/S0308210504000721
Jérôme Coville and Louis Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal. 60 (2005), no. 5, 797–819. MR 2113158, DOI 10.1016/j.na.2003.10.030
R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), 355–369.
F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math. 52 (1999), no. 10, 1255–1276. MR 1699968, DOI 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.3.CO;2-N
A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Étude de l’équation de la chaleur de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh. 1 (1937), 1–25.
Wan-Tong Li, Yu-Juan Sun, and Zhi-Cheng Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2302–2313. MR 2661901, DOI 10.1016/j.nonrwa.2009.07.005
Antoine Mellet, James Nolen, Jean-Michel Roquejoffre, and Lenya Ryzhik, Stability of generalized transition fronts, Comm. Partial Differential Equations 34 (2009), no. 4-6, 521–552. MR 2530708, DOI 10.1080/03605300902768677
Antoine Mellet, Jean-Michel Roquejoffre, and Yannick Sire, Generalized fronts for one-dimensional reaction-diffusion equations, Discrete Contin. Dyn. Syst. 26 (2010), no. 1, 303–312. MR 2552789, DOI 10.3934/dcds.2010.26.303
Antoine Mellet, Jean-Michel Roquejoffre, and Yannick Sire, Existence and asymptotics of fronts in non local combustion models, Commun. Math. Sci. 12 (2014), no. 1, 1–11. MR 3100891, DOI 10.4310/CMS.2014.v12.n1.a1
James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik, and Andrej Zlatoš, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal. 203 (2012), no. 1, 217–246. MR 2864411, DOI 10.1007/s00205-011-0449-4
James Nolen and Lenya Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 3, 1021–1047. MR 2526414, DOI 10.1016/j.anihpc.2009.02.003
Tianyu Tao, Beite Zhu, and Andrej Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero, Nonlinearity 27 (2014), no. 9, 2409–2416. MR 3266860, DOI 10.1088/0951-7715/27/9/2409
K\B{o}hei Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ. 18 (1978), no. 3, 453–508. MR 509494, DOI 10.1215/kjm/1250522506
S. Vakulenko and V. Volpert, Generalized travelling waves for perturbed monotone reaction-diffusion systems, Nonlinear Anal. 46 (2001), no. 6, Ser. A: Theory Methods, 757–776. MR 1859795, DOI 10.1016/S0362-546X(00)00130-9
Jack Xin, Front propagation in heterogeneous media, SIAM Rev. 42 (2000), no. 2, 161–230. MR 1778352, DOI 10.1137/S0036144599364296
Andrej Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl. (9) 98 (2012), no. 1, 89–102. MR 2935371, DOI 10.1016/j.matpur.2011.11.007
Andrej Zlatoš, Generalized traveling waves in disordered media: existence, uniqueness, and stability, Arch. Ration. Mech. Anal. 208 (2013), no. 2, 447–480. MR 3035984, DOI 10.1007/s00205-012-0600-x