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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 and 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, Arch. Rational Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037. [2] H. Berestycki, J. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., DOI: 10.1007/s00285-015-0911-2. [3] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, 101-123, Contemp. Math., 446, Amer. Math. Soc., Providence, RI, 2007. doi: 10.1090/conm/446/08627. [4] H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648. doi: 10.1002/cpa.21389. [5] 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. [6] X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258. doi: 10.1137/050627824. [7] X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569. doi: 10.1006/jdeq.2001.4153. [8] X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146. doi: 10.1007/s00208-003-0414-0. [9] 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. [10] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721. [11] J. Coville, J. 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. [12] S.-C. Fu, J.-S. Guo and S.-Y Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations, Nonlinear Anal., 48 (2002), 1137-1149. doi: 10.1016/S0362-546X(00)00242-X. [13] J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525. doi: 10.1007/s00208-005-0729-0. [14] J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations, 246 (2009), 3818-3833. doi: 10.1016/j.jde.2009.03.010. [15] W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equation, Comm. Appl. Nonlinear Anal., 1 (1994), 23-46. [16] T. Lim and A. Zlatoš, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., DOI: http://dx.doi.org/10.1090/tran/6602. [17] 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. [18] J. Nolen, J.-M. Roquejoffre, L. 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. [19] N. Rawal, W. 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. [20] 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. [21] W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060. doi: 10.1007/s10884-004-7832-x. [22] W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, J. Dynam. Differential Equations, 23 (2011), 1-44. doi: 10.1007/s10884-010-9200-3. [23] W. Shen and Z. Shen, Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity, Discrete Contin. Dyn. Syst. A, accepted. http://arxiv.org/abs/1410.4611. [24] W. Shen and Z. Shen, Regularity and stability of transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity, http://arxiv.org/abs/1501.02029. [25] W. Shen and Z. Shen, Regularity of transition fronts in nonlocal dispersal evolution equations, J. Dynam. Differential Equations, DOI: 10.1007/s10884-016-9528-4. [26] B. Shorrocks and I. Swingland, Living in a Patch Environment, Oxford Univ. Press, New York, 1990. [27] 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. [28] 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. [29] W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101. [30] T. Tao, B. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero, Nonlinearity, 27 (2014), 2409-2416. doi: 10.1088/0951-7715/27/9/2409. [31] K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. [32] J. Wu and X. Zou, Asymptotic and periodic boundary values problems of mixed PDEs and wave solutions of lattice differential equations, J. Differential Equations, 135 (1997), 315-357. doi: 10.1006/jdeq.1996.3232. [33] 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. [34] B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62. doi: 10.1006/jdeq.1993.1082.

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, Arch. Rational Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037. [2] H. Berestycki, J. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., DOI: 10.1007/s00285-015-0911-2. [3] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, 101-123, Contemp. Math., 446, Amer. Math. Soc., Providence, RI, 2007. doi: 10.1090/conm/446/08627. [4] H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648. doi: 10.1002/cpa.21389. [5] 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. [6] X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258. doi: 10.1137/050627824. [7] X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569. doi: 10.1006/jdeq.2001.4153. [8] X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146. doi: 10.1007/s00208-003-0414-0. [9] 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. [10] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721. [11] J. Coville, J. 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. [12] S.-C. Fu, J.-S. Guo and S.-Y Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations, Nonlinear Anal., 48 (2002), 1137-1149. doi: 10.1016/S0362-546X(00)00242-X. [13] J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525. doi: 10.1007/s00208-005-0729-0. [14] J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations, 246 (2009), 3818-3833. doi: 10.1016/j.jde.2009.03.010. [15] W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equation, Comm. Appl. Nonlinear Anal., 1 (1994), 23-46. [16] T. Lim and A. Zlatoš, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., DOI: http://dx.doi.org/10.1090/tran/6602. [17] 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. [18] J. Nolen, J.-M. Roquejoffre, L. 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. [19] N. Rawal, W. 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. [20] 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. [21] W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060. doi: 10.1007/s10884-004-7832-x. [22] W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, J. Dynam. Differential Equations, 23 (2011), 1-44. doi: 10.1007/s10884-010-9200-3. [23] W. Shen and Z. Shen, Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity, Discrete Contin. Dyn. Syst. A, accepted. http://arxiv.org/abs/1410.4611. [24] W. Shen and Z. Shen, Regularity and stability of transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity, http://arxiv.org/abs/1501.02029. [25] W. Shen and Z. Shen, Regularity of transition fronts in nonlocal dispersal evolution equations, J. Dynam. Differential Equations, DOI: 10.1007/s10884-016-9528-4. [26] B. Shorrocks and I. Swingland, Living in a Patch Environment, Oxford Univ. Press, New York, 1990. [27] 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. [28] 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. [29] W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101. [30] T. Tao, B. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero, Nonlinearity, 27 (2014), 2409-2416. doi: 10.1088/0951-7715/27/9/2409. [31] K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. [32] J. Wu and X. Zou, Asymptotic and periodic boundary values problems of mixed PDEs and wave solutions of lattice differential equations, J. Differential Equations, 135 (1997), 315-357. doi: 10.1006/jdeq.1996.3232. [33] 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. [34] B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62. doi: 10.1006/jdeq.1993.1082.
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