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July  2016, 15(4): 1193-1213. doi: 10.3934/cpaa.2016.15.1193

Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media

1. 

Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849

2. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada

Received  June 2015 Revised  January 2016 Published  April 2016

The present paper is devoted to the study of transition fronts of nonlocal Fisher-KPP equations in time heterogeneous media. We first construct transition fronts with exact decaying rates as the space variable tends to infinity and with prescribed interface location functions, which are natural generalizations of front location functions in homogeneous media. Then, by the general results on space regularity of transition fronts of nonlocal evolution equations proven in the authors' earlier work ([25]), these transition fronts are continuously differentiable in space. We show that their space partial derivatives have exact decaying rates as the space variable tends to infinity. Finally, we study the asymptotic stability of transition fronts. It is shown that transition fronts attract those solutions whose initial data decays as fast as transition fronts near infinity and essentially above zero near negative infinity.
Citation: Wenxian Shen, Zhongwei Shen. Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1193-1213. doi: 10.3934/cpaa.2016.15.1193
References:
[1]

P.W. Bates, P.C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, \emph{Arch. Rational Mech. Anal.}, 138 (1997), 105. doi: 10.1007/s002050050037. Google Scholar

[2]

H. Berestycki, J. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion,, \emph{J. Math. Biol.}, (): 00285. Google Scholar

[3]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, 101-123,, \emph{Contemp. Math.}, (2007). doi: 10.1090/conm/446/08627. Google Scholar

[4]

H. Berestycki and F. Hamel, Generalized transition waves and their properties,, \emph{Comm. Pure Appl. Math.}, 65 (2012), 592. doi: 10.1002/cpa.21389. Google Scholar

[5]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, \emph{Proc. Amer. Math. Soc.}, 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar

[6]

X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices,, \emph{SIAM J. Math. Anal.}, 38 (2006), 233. doi: 10.1137/050627824. Google Scholar

[7]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations,, \emph{J. Differential Equations}, 184 (2002), 549. doi: 10.1006/jdeq.2001.4153. Google Scholar

[8]

X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics,, \emph{Math. Ann.}, 326 (2003), 123. doi: 10.1007/s00208-003-0414-0. Google Scholar

[9]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations,, \emph{Nonlinear Anal.}, 60 (2005), 797. doi: 10.1016/j.na.2003.10.030. Google Scholar

[10]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 137 (2007), 727. doi: 10.1017/S0308210504000721. Google Scholar

[11]

J. Coville, J. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 30 (2013), 179. doi: 10.1016/j.anihpc.2012.07.005. Google Scholar

[12]

S.-C. Fu, J.-S. Guo and S.-Y Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations,, \emph{Nonlinear Anal.}, 48 (2002), 1137. doi: 10.1016/S0362-546X(00)00242-X. Google Scholar

[13]

J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations,, \emph{Math. Ann.}, 335 (2006), 489. doi: 10.1007/s00208-005-0729-0. Google Scholar

[14]

J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system,, \emph{J. Differential Equations}, 246 (2009), 3818. doi: 10.1016/j.jde.2009.03.010. Google Scholar

[15]

W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equation,, \emph{Comm. Appl. Nonlinear Anal.}, 1 (1994), 23. Google Scholar

[16]

T. Lim and A. Zlatoš, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion,, \emph{Trans. Amer. Math. Soc.}, (). Google Scholar

[17]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations,, \emph{J. Math. Pures Appl. (9)}, 98 (2012), 633. doi: 10.1016/j.matpur.2012.05.005. Google Scholar

[18]

J. Nolen, J.-M. Roquejoffre, L. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts,, \emph{Arch. Ration. Mech. Anal.}, 203 (2012), 217. doi: 10.1007/s00205-011-0449-4. Google Scholar

[19]

N. Rawal, W. Shen and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats,, \emph{Discrete Contin. Dyn. Syst.}, 35 (2015), 1609. doi: 10.3934/dcds.2015.35.1609. Google Scholar

[20]

K. Schumacher, Traveling-front solutions for integro-differential equations. I,, \emph{J. Reine Angew. Math.}, 316 (1980), 54. doi: 10.1515/crll.1980.316.54. Google Scholar

[21]

W. Shen, Traveling waves in diffusive random media,, \emph{J. Dynam. Differential Equations}, 16 (2004), 1011. doi: 10.1007/s10884-004-7832-x. Google Scholar

[22]

W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations,, \emph{J. Dynam. Differential Equations}, 23 (2011), 1. doi: 10.1007/s10884-010-9200-3. Google Scholar

[23]

W. Shen and Z. Shen, Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity,, \emph{Discrete Contin. Dyn. Syst. A}, (). Google Scholar

[24]

W. Shen and Z. Shen, Regularity and stability of transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity,, \url{http://arxiv.org/abs/1501.02029}., (). Google Scholar

[25]

W. Shen and Z. Shen, Regularity of transition fronts in nonlocal dispersal evolution equations,, \emph{J. Dynam. Differential Equations}, (): 10884. Google Scholar

[26]

B. Shorrocks and I. Swingland, Living in a Patch Environment,, Oxford Univ. Press, (1990). Google Scholar

[27]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, \emph{J. Differential Equations}, 249 (2010), 747. doi: 10.1016/j.jde.2010.04.012. Google Scholar

[28]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats,, \emph{Proc. Amer. Math. Soc.}, 140 (2012), 1681. doi: 10.1090/S0002-9939-2011-11011-6. Google Scholar

[29]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations,, \emph{Comm. Appl. Nonlinear Anal.}, 19 (2012), 73. Google Scholar

[30]

T. Tao, B. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero,, \emph{Nonlinearity}, 27 (2014), 2409. doi: 10.1088/0951-7715/27/9/2409. Google Scholar

[31]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time,, \emph{J. Math. Kyoto Univ.}, 18 (1978), 453. Google Scholar

[32]

J. Wu and X. Zou, Asymptotic and periodic boundary values problems of mixed PDEs and wave solutions of lattice differential equations,, \emph{J. Differential Equations}, 135 (1997), 315. doi: 10.1006/jdeq.1996.3232. Google Scholar

[33]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations,, \emph{J. Math. Pures Appl. (9)}, 98 (2012), 89. doi: 10.1016/j.matpur.2011.11.007. Google Scholar

[34]

B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation,, \emph{J. Differential Equations}, 105 (1993), 46. doi: 10.1006/jdeq.1993.1082. Google Scholar

show all references

References:
[1]

P.W. Bates, P.C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, \emph{Arch. Rational Mech. Anal.}, 138 (1997), 105. doi: 10.1007/s002050050037. Google Scholar

[2]

H. Berestycki, J. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion,, \emph{J. Math. Biol.}, (): 00285. Google Scholar

[3]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, 101-123,, \emph{Contemp. Math.}, (2007). doi: 10.1090/conm/446/08627. Google Scholar

[4]

H. Berestycki and F. Hamel, Generalized transition waves and their properties,, \emph{Comm. Pure Appl. Math.}, 65 (2012), 592. doi: 10.1002/cpa.21389. Google Scholar

[5]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, \emph{Proc. Amer. Math. Soc.}, 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar

[6]

X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices,, \emph{SIAM J. Math. Anal.}, 38 (2006), 233. doi: 10.1137/050627824. Google Scholar

[7]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations,, \emph{J. Differential Equations}, 184 (2002), 549. doi: 10.1006/jdeq.2001.4153. Google Scholar

[8]

X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics,, \emph{Math. Ann.}, 326 (2003), 123. doi: 10.1007/s00208-003-0414-0. Google Scholar

[9]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations,, \emph{Nonlinear Anal.}, 60 (2005), 797. doi: 10.1016/j.na.2003.10.030. Google Scholar

[10]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 137 (2007), 727. doi: 10.1017/S0308210504000721. Google Scholar

[11]

J. Coville, J. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 30 (2013), 179. doi: 10.1016/j.anihpc.2012.07.005. Google Scholar

[12]

S.-C. Fu, J.-S. Guo and S.-Y Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations,, \emph{Nonlinear Anal.}, 48 (2002), 1137. doi: 10.1016/S0362-546X(00)00242-X. Google Scholar

[13]

J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations,, \emph{Math. Ann.}, 335 (2006), 489. doi: 10.1007/s00208-005-0729-0. Google Scholar

[14]

J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system,, \emph{J. Differential Equations}, 246 (2009), 3818. doi: 10.1016/j.jde.2009.03.010. Google Scholar

[15]

W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equation,, \emph{Comm. Appl. Nonlinear Anal.}, 1 (1994), 23. Google Scholar

[16]

T. Lim and A. Zlatoš, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion,, \emph{Trans. Amer. Math. Soc.}, (). Google Scholar

[17]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations,, \emph{J. Math. Pures Appl. (9)}, 98 (2012), 633. doi: 10.1016/j.matpur.2012.05.005. Google Scholar

[18]

J. Nolen, J.-M. Roquejoffre, L. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts,, \emph{Arch. Ration. Mech. Anal.}, 203 (2012), 217. doi: 10.1007/s00205-011-0449-4. Google Scholar

[19]

N. Rawal, W. Shen and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats,, \emph{Discrete Contin. Dyn. Syst.}, 35 (2015), 1609. doi: 10.3934/dcds.2015.35.1609. Google Scholar

[20]

K. Schumacher, Traveling-front solutions for integro-differential equations. I,, \emph{J. Reine Angew. Math.}, 316 (1980), 54. doi: 10.1515/crll.1980.316.54. Google Scholar

[21]

W. Shen, Traveling waves in diffusive random media,, \emph{J. Dynam. Differential Equations}, 16 (2004), 1011. doi: 10.1007/s10884-004-7832-x. Google Scholar

[22]

W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations,, \emph{J. Dynam. Differential Equations}, 23 (2011), 1. doi: 10.1007/s10884-010-9200-3. Google Scholar

[23]

W. Shen and Z. Shen, Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity,, \emph{Discrete Contin. Dyn. Syst. A}, (). Google Scholar

[24]

W. Shen and Z. Shen, Regularity and stability of transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity,, \url{http://arxiv.org/abs/1501.02029}., (). Google Scholar

[25]

W. Shen and Z. Shen, Regularity of transition fronts in nonlocal dispersal evolution equations,, \emph{J. Dynam. Differential Equations}, (): 10884. Google Scholar

[26]

B. Shorrocks and I. Swingland, Living in a Patch Environment,, Oxford Univ. Press, (1990). Google Scholar

[27]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, \emph{J. Differential Equations}, 249 (2010), 747. doi: 10.1016/j.jde.2010.04.012. Google Scholar

[28]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats,, \emph{Proc. Amer. Math. Soc.}, 140 (2012), 1681. doi: 10.1090/S0002-9939-2011-11011-6. Google Scholar

[29]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations,, \emph{Comm. Appl. Nonlinear Anal.}, 19 (2012), 73. Google Scholar

[30]

T. Tao, B. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero,, \emph{Nonlinearity}, 27 (2014), 2409. doi: 10.1088/0951-7715/27/9/2409. Google Scholar

[31]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time,, \emph{J. Math. Kyoto Univ.}, 18 (1978), 453. Google Scholar

[32]

J. Wu and X. Zou, Asymptotic and periodic boundary values problems of mixed PDEs and wave solutions of lattice differential equations,, \emph{J. Differential Equations}, 135 (1997), 315. doi: 10.1006/jdeq.1996.3232. Google Scholar

[33]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations,, \emph{J. Math. Pures Appl. (9)}, 98 (2012), 89. doi: 10.1016/j.matpur.2011.11.007. Google Scholar

[34]

B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation,, \emph{J. Differential Equations}, 105 (1993), 46. doi: 10.1006/jdeq.1993.1082. Google Scholar

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