February  2017, 37(2): 1013-1037. doi: 10.3934/dcds.2017042

Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity

1. 

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

2. 

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

* Corresponding author

Received  December 2014 Revised  July 2015 Published  November 2016

The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equations with time heterogeneous nonlinearity of ignition type. It is proven that such an equation admits space monotone transition fronts with finite speed and space regularity in the sense of uniform Lipschitz continuity. Our approach is first constructing a sequence of approximating front-like solutions and then proving that the approximating solutions converge to a transition front. We take advantage of the idea of modified interface location, which allows us to characterize the finite speed of approximating solutions in the absence of space regularity, and leads directly to uniform exponential decaying estimates.

Citation: Wenxian Shen, Zhongwei Shen. Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1013-1037. doi: 10.3934/dcds.2017042
References:
[1]

N. AlikakosP. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805.  doi: 10.1090/S0002-9947-99-02134-0.  Google Scholar

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H. BerestyckiJ. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., 72 (2016), 1693-1745.  doi: 10.1007/s00285-015-0911-2.  Google Scholar

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H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.  doi: 10.1002/cpa.3022.  Google Scholar

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H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Amer. Math. Soc., 446 (2007), 101-123.  doi: 10.1090/conm/446/08627.  Google Scholar

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H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.  Google Scholar

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J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

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X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.   Google Scholar

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J. Coville, Équations de réaction-diffusion non-locale, PhD thesis. http://hal.upmc.fr/tel-00004313/document Google Scholar

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J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.  Google Scholar

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J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.  Google Scholar

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J. CovilleJ. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

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W. Ding, F. Hamel and X. -Q. Zhao, Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat, Indiana Univ. Math. J. , in press. Google Scholar

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W. Ding and X. Liang, Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media, SIAM J. Math. Anal., 47 (2015), 855-896.  doi: 10.1137/140958141.  Google Scholar

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P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.   Google Scholar

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P. C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, Arch. Rational Mech. Anal., 75 (1980/81), 281-314.  doi: 10.1007/BF00256381.  Google Scholar

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V. HutsonW. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math., 38 (2008), 1147-1175.  doi: 10.1216/RMJ-2008-38-4-1147.  Google Scholar

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Y. Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka J. Math., 13 (1976), 11-66.   Google Scholar

[23]

A. KolmogorovI. Petrowsky and N. Piscunov, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem, Dynamics of Curved Fronts, (1988), 105-130.  doi: 10.1016/B978-0-08-092523-3.50014-9.  Google Scholar

[24]

L. Kong and W. Shen, Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity, J. Dynam. Differential Equations, 26 (2014), 181-215.  doi: 10.1007/s10884-014-9351-8.  Google Scholar

[25]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[26]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[27]

T. Lim and A. Zlatoš, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615-8631.  doi: 10.1090/tran/6602.  Google Scholar

[28]

A. MelletJ.-M. Roquejoffre and Y. Sire, Generalized fronts for one-dimensional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 26 (2010), 303-312.  doi: 10.3934/dcds.2010.26.303.  Google Scholar

[29]

A. MelletJ. NolenJ.-M. Roquejoffre and L. Ryzhik, Stability of generalized transition fronts, Comm. Partial Differential Equations, 34 (2009), 521-552.  doi: 10.1080/03605300902768677.  Google Scholar

[30]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl(9), 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.  Google Scholar

[31]

G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation, J. Differential Equations, 249 (2010), 1288-1304.  doi: 10.1016/j.jde.2010.05.007.  Google Scholar

[32]

G. Nadin, Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 841-873.  doi: 10.1016/j.anihpc.2014.03.007.  Google Scholar

[33]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl(9), 98 (2012), 633-653.  doi: 10.1016/j.matpur.2012.05.005.  Google Scholar

[34]

G. Nadin and L. Rossi, Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coeffcients, Anal. PDE, 8 (2015), 1351-1377.  doi: 10.2140/apde.2015.8.1351.  Google Scholar

[35]

J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1021-1047.  doi: 10.1016/j.anihpc.2009.02.003.  Google Scholar

[36]

J. NolenJ.-M. RoquejoffreL. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal., 203 (2012), 217-246.  doi: 10.1007/s00205-011-0449-4.  Google Scholar

[37]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York Berlin Heidelberg Tokyo, 1983. Google Scholar

[38]

N. RawalW. Shen and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst., 35 (2015), 1609-1640.  doi: 10.3934/dcds.2015.35.1609.  Google Scholar

[39]

K. Schumacher, Traveling-front solutions for integro-differential equations. I, J. Reine Angew. Math., 316 (1980), 54-70.  doi: 10.1515/crll.1980.316.54.  Google Scholar

[40]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. I. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54.  doi: 10.1006/jdeq.1999.3651.  Google Scholar

[41]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. II. Existence, J. Differential Equations, 159 (1999), 55-101.  doi: 10.1006/jdeq.1999.3652.  Google Scholar

[42]

W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.  Google Scholar

[43]

W. Shen, Traveling waves in time dependent bistable equations, Differential Integral Equations, 19 (2006), 241-278.   Google Scholar

[44]

W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, . Dynam. Differential Equations, 23 (2011), 1-44.  doi: 10.1007/s10884-010-9200-3.  Google Scholar

[45]

W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93.   Google Scholar

[46]

W. Shen and Z. Shen, Transition fronts in time heterogeneous and random media of ignition type, Journal of Differential Equations, 2016, http://arxiv.org/abs/1407.7579. Google Scholar

[47]

W. Shen and Z. Shen, Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type, Trans. Amer. Math. Soc., (2016).  doi: 10.1090/tran/6726.  Google Scholar

[48]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[49]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar

[50]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101.   Google Scholar

[51]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.   Google Scholar

[52]

H. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.  Google Scholar

[53]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

[54]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl(9), 98 (2012), 89-102.  doi: 10.1016/j.matpur.2011.11.007.  Google Scholar

[55]

A. Zlatoš, Generalized traveling waves in disordered media: Existence, uniqueness, and stability, Arch. Ration. Mech. Anal., 208 (2013), 447-480.  doi: 10.1007/s00205-012-0600-x.  Google Scholar

show all references

References:
[1]

N. AlikakosP. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805.  doi: 10.1090/S0002-9947-99-02134-0.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Lecture Notes in Math. , Springer, Berlin, 446 (1975), 5-49. Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[4]

P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.  Google Scholar

[5]

H. BerestyckiJ. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., 72 (2016), 1693-1745.  doi: 10.1007/s00285-015-0911-2.  Google Scholar

[6]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.  doi: 10.1002/cpa.3022.  Google Scholar

[7]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Amer. Math. Soc., 446 (2007), 101-123.  doi: 10.1090/conm/446/08627.  Google Scholar

[8]

H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.  Google Scholar

[9]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[10]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.   Google Scholar

[11]

J. Coville, Équations de réaction-diffusion non-locale, PhD thesis. http://hal.upmc.fr/tel-00004313/document Google Scholar

[12]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.  Google Scholar

[13]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.  Google Scholar

[14]

J. CovilleJ. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[15]

W. Ding, F. Hamel and X. -Q. Zhao, Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat, Indiana Univ. Math. J. , in press. Google Scholar

[16]

W. Ding and X. Liang, Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media, SIAM J. Math. Anal., 47 (2015), 855-896.  doi: 10.1137/140958141.  Google Scholar

[17]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 335-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[18]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153–191, Springer, Berlin, 2003. Google Scholar

[19]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.   Google Scholar

[20]

P. C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, Arch. Rational Mech. Anal., 75 (1980/81), 281-314.  doi: 10.1007/BF00256381.  Google Scholar

[21]

V. HutsonW. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math., 38 (2008), 1147-1175.  doi: 10.1216/RMJ-2008-38-4-1147.  Google Scholar

[22]

Y. Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka J. Math., 13 (1976), 11-66.   Google Scholar

[23]

A. KolmogorovI. Petrowsky and N. Piscunov, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem, Dynamics of Curved Fronts, (1988), 105-130.  doi: 10.1016/B978-0-08-092523-3.50014-9.  Google Scholar

[24]

L. Kong and W. Shen, Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity, J. Dynam. Differential Equations, 26 (2014), 181-215.  doi: 10.1007/s10884-014-9351-8.  Google Scholar

[25]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[26]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[27]

T. Lim and A. Zlatoš, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615-8631.  doi: 10.1090/tran/6602.  Google Scholar

[28]

A. MelletJ.-M. Roquejoffre and Y. Sire, Generalized fronts for one-dimensional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 26 (2010), 303-312.  doi: 10.3934/dcds.2010.26.303.  Google Scholar

[29]

A. MelletJ. NolenJ.-M. Roquejoffre and L. Ryzhik, Stability of generalized transition fronts, Comm. Partial Differential Equations, 34 (2009), 521-552.  doi: 10.1080/03605300902768677.  Google Scholar

[30]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl(9), 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.  Google Scholar

[31]

G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation, J. Differential Equations, 249 (2010), 1288-1304.  doi: 10.1016/j.jde.2010.05.007.  Google Scholar

[32]

G. Nadin, Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 841-873.  doi: 10.1016/j.anihpc.2014.03.007.  Google Scholar

[33]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl(9), 98 (2012), 633-653.  doi: 10.1016/j.matpur.2012.05.005.  Google Scholar

[34]

G. Nadin and L. Rossi, Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coeffcients, Anal. PDE, 8 (2015), 1351-1377.  doi: 10.2140/apde.2015.8.1351.  Google Scholar

[35]

J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1021-1047.  doi: 10.1016/j.anihpc.2009.02.003.  Google Scholar

[36]

J. NolenJ.-M. RoquejoffreL. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal., 203 (2012), 217-246.  doi: 10.1007/s00205-011-0449-4.  Google Scholar

[37]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York Berlin Heidelberg Tokyo, 1983. Google Scholar

[38]

N. RawalW. Shen and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst., 35 (2015), 1609-1640.  doi: 10.3934/dcds.2015.35.1609.  Google Scholar

[39]

K. Schumacher, Traveling-front solutions for integro-differential equations. I, J. Reine Angew. Math., 316 (1980), 54-70.  doi: 10.1515/crll.1980.316.54.  Google Scholar

[40]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. I. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54.  doi: 10.1006/jdeq.1999.3651.  Google Scholar

[41]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. II. Existence, J. Differential Equations, 159 (1999), 55-101.  doi: 10.1006/jdeq.1999.3652.  Google Scholar

[42]

W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.  Google Scholar

[43]

W. Shen, Traveling waves in time dependent bistable equations, Differential Integral Equations, 19 (2006), 241-278.   Google Scholar

[44]

W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, . Dynam. Differential Equations, 23 (2011), 1-44.  doi: 10.1007/s10884-010-9200-3.  Google Scholar

[45]

W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93.   Google Scholar

[46]

W. Shen and Z. Shen, Transition fronts in time heterogeneous and random media of ignition type, Journal of Differential Equations, 2016, http://arxiv.org/abs/1407.7579. Google Scholar

[47]

W. Shen and Z. Shen, Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type, Trans. Amer. Math. Soc., (2016).  doi: 10.1090/tran/6726.  Google Scholar

[48]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[49]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar

[50]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101.   Google Scholar

[51]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.   Google Scholar

[52]

H. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.  Google Scholar

[53]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

[54]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl(9), 98 (2012), 89-102.  doi: 10.1016/j.matpur.2011.11.007.  Google Scholar

[55]

A. Zlatoš, Generalized traveling waves in disordered media: Existence, uniqueness, and stability, Arch. Ration. Mech. Anal., 208 (2013), 447-480.  doi: 10.1007/s00205-012-0600-x.  Google Scholar

Figure 1.  Modified Interface Location
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