February  2021, 26(2): 1031-1060. doi: 10.3934/dcdsb.2020152

Entire solutions originating from monotone fronts for nonlocal dispersal equations with bistable nonlinearity

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P.R. China

2. 

School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, P.R. China

3. 

School of Science, Chang'an University, Xi'an, Shaanxi 710064, P.R. China

* Corresponding author: Wan-Tong Li

Received  July 2019 Revised  November 2019 Published  February 2021 Early access  May 2020

This paper mainly focuses on the entire solutions of nonlocal dispersal equations with bistable nonlinearity. Under certain assumptions of wave speed, firstly constructing appropriate super- and sub-solutions and applying corresponding comparison principle, we established the existence and related properties of entire solutions formed by the collision of three and four traveling wave solutions. Then by introducing the definition of terminated sequence, it is proved that there has no entire solutions formed by $ k $ traveling wave solutions that collide with each other as long as $ k\geq5 $. Finally, based on the classical weighted energy approach, we obtain the global exponentially stability of the entire solutions in some weighted space.

Citation: Fang-Di Dong, Wan-Tong Li, Shi-Liang Wu, Li Zhang. Entire solutions originating from monotone fronts for nonlocal dispersal equations with bistable nonlinearity. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1031-1060. doi: 10.3934/dcdsb.2020152
References:
[1]

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.

[2]

P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.

[3]

J. F. CaoY. DuF. Li and W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.

[4]

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.

[5]

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

[6]

X. Chen and J. S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028.

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Y. Y. Chen, Entire solutions originating from three fronts for a discrete diffusive equation, Tamkang Journal of Mathematics, 48 (2017), 215-226.  doi: 10.5556/j.tkjm.48.2017.2442.

[8]

Y. Y. ChenJ. S. GuoN. Ninomiya and C. H. Yao, Entire solutions originating from monotone fronts to the Allen-Cahn equation, Phys. D, 378/379 (2018), 1-19.  doi: 10.1016/j.physd.2018.04.003.

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C. CortazarM. ElguetaJ. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002.

[10]

C. CortazarM. ElguetaJ. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8.

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J. Coville, Traveling Fronts in Asymmetric Nonlocal Reaction Diffusion Equation: The Bistable and Ignition Case, Prépublication du CMM, Hal-00696208.

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J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.

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

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E. C. M. Crooks and J. C. Tsai, Front-like entire solutions for equations with convection, J. Differential Equations, 253 (2012), 1206-1249.  doi: 10.1016/j.jde.2012.04.022.

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F. D. DongW. T. Li and J. B. Wang, Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application, Discrete Contin. Dyn. Syst., 37 (2017), 6291-6318.  doi: 10.3934/dcds.2017272.

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J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.

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F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:103.0.CO;2-W.

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F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^{N}$, Arch. Rational Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.

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V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.

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V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.

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L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013.

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W. T. LiN. W. Liu and Z. C. Wang, Entire solutions in reaction-advection-diffusion equations in cylinders, J. Math. Pures Appl., 90 (2008), 492-504.  doi: 10.1016/j.matpur.2008.07.002.

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W. T. LiY. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.

[26]

W. T. LiJ. B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006.

[27]

W. T. LiZ. C. Wang and J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.  doi: 10.1016/j.jde.2008.03.023.

[28]

W. T. LiL. Zhang and G. B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.

[29]

Y. LiW. T. Li and G. B. Zhang, Stability and uniqueness of traveling waves of a nonlocal dispersal SIR epidemic model, Dyn. Partial Differ. Equ., 14 (2017), 87-123.  doi: 10.4310/DPDE.2017.v14.n2.a1.

[30]

C. K. LinC. T. LinY. P. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084.  doi: 10.1137/120904391.

[31]

N. W. LiuW. T. Li and Z. C. Wang, Entire solutions of reaction-advection-diffusion equations with bistable nonlinearity in cylinders, J. Differential Equations, 246 (2009), 4249-4267.  doi: 10.1016/j.jde.2008.12.005.

[32]

M. MeiC. K. LinC. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529.  doi: 10.1016/j.jde.2008.12.020.

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M. MeiC. K. LinC. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity, J. Differential Equations, 247 (2009), 495-510.  doi: 10.1016/j.jde.2008.12.026.

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M. MeiC. Ou and X. Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790.  doi: 10.1137/090776342.

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M. Mei and J. W. H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551-568.  doi: 10.1017/S0308210506000333.

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M. MeiJ. W. H. SoM. Y. Li and S. S. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594.  doi: 10.1017/S0308210500003358.

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Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.

[38]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715.

[39]

S. PanW. T. Li and G. Lin, Travelling wave fronts in nonlocal reaction–diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.  doi: 10.1007/s00033-007-7005-y.

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K. Schumacher, Travelling-front solutions for integro-differential equations, I, J. Reine Angew. Math., 1980 (2009), 54-70.  doi: 10.1515/crll.1980.316.54.

[41]

H. L. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.

[42]

Y. J. SunW. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.  doi: 10.1016/j.jde.2011.04.020.

[43]

Y. J. SunL. ZhangW. T. Li and Z. C. Wang, Entire solutions in nonlocal monostable equations: Asymmetric case, Comm. Pure Appl. Anal., 18 (2019), 1049-1072.  doi: 10.3934/cpaa.2019051.

[44]

M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005.

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Z. C. WangW. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1.

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Z. C. WangW. T. Li and S. Ruan, Entire solutions in lattice delayed differential equations with nonlocal interaction: bistable cases, Math. Model. Nat. Phenom., 8 (2013), 78-103.  doi: 10.1051/mmnp/20138307.

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Z. C. WangW. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.

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S. L. Wu and C. H. Hsu, Entire solutions with merging fronts to a bistable periodic lattice dynamical system, Discrete Contin. Dyn. Syst., 36 (2016), 2329-2346.  doi: 10.3934/dcds.2016.36.2329.

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T. XuS. JiR. HuangM. Mei and J. Yin, Theoretical and numerical studies on global stability of traveling waves with oscillations for time-delayed nonlocal dispersion equations, Int. J. Numer. Anal. Model., 17 (2020), 68-86. 

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show all references

References:
[1]

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.

[2]

P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.

[3]

J. F. CaoY. DuF. Li and W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.

[4]

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.

[5]

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

[6]

X. Chen and J. S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028.

[7]

Y. Y. Chen, Entire solutions originating from three fronts for a discrete diffusive equation, Tamkang Journal of Mathematics, 48 (2017), 215-226.  doi: 10.5556/j.tkjm.48.2017.2442.

[8]

Y. Y. ChenJ. S. GuoN. Ninomiya and C. H. Yao, Entire solutions originating from monotone fronts to the Allen-Cahn equation, Phys. D, 378/379 (2018), 1-19.  doi: 10.1016/j.physd.2018.04.003.

[9]

C. CortazarM. ElguetaJ. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002.

[10]

C. CortazarM. ElguetaJ. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8.

[11]

J. Coville, Traveling Fronts in Asymmetric Nonlocal Reaction Diffusion Equation: The Bistable and Ignition Case, Prépublication du CMM, Hal-00696208.

[12]

J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.

[13]

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

[14]

E. C. M. Crooks and J. C. Tsai, Front-like entire solutions for equations with convection, J. Differential Equations, 253 (2012), 1206-1249.  doi: 10.1016/j.jde.2012.04.022.

[15]

F. D. DongW. T. Li and J. B. Wang, Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application, Discrete Contin. Dyn. Syst., 37 (2017), 6291-6318.  doi: 10.3934/dcds.2017272.

[16]

J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.

[17]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:103.0.CO;2-W.

[18]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^{N}$, Arch. Rational Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.

[19]

R. HuangM. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Contin. Dyn. Syst., 32 (2012), 3621-3649.  doi: 10.3934/dcds.2012.32.3621.

[20]

R. HuangM. MeiK. J. Zhang and Q. F. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations, Discrete Contin. Dyn. Syst., 36 (2016), 1331-1353.  doi: 10.3934/dcds.2016.36.1331.

[21]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.

[22]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.

[23]

L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013.

[24]

W. T. LiN. W. Liu and Z. C. Wang, Entire solutions in reaction-advection-diffusion equations in cylinders, J. Math. Pures Appl., 90 (2008), 492-504.  doi: 10.1016/j.matpur.2008.07.002.

[25]

W. T. LiY. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.

[26]

W. T. LiJ. B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006.

[27]

W. T. LiZ. C. Wang and J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.  doi: 10.1016/j.jde.2008.03.023.

[28]

W. T. LiL. Zhang and G. B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.

[29]

Y. LiW. T. Li and G. B. Zhang, Stability and uniqueness of traveling waves of a nonlocal dispersal SIR epidemic model, Dyn. Partial Differ. Equ., 14 (2017), 87-123.  doi: 10.4310/DPDE.2017.v14.n2.a1.

[30]

C. K. LinC. T. LinY. P. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084.  doi: 10.1137/120904391.

[31]

N. W. LiuW. T. Li and Z. C. Wang, Entire solutions of reaction-advection-diffusion equations with bistable nonlinearity in cylinders, J. Differential Equations, 246 (2009), 4249-4267.  doi: 10.1016/j.jde.2008.12.005.

[32]

M. MeiC. K. LinC. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529.  doi: 10.1016/j.jde.2008.12.020.

[33]

M. MeiC. K. LinC. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity, J. Differential Equations, 247 (2009), 495-510.  doi: 10.1016/j.jde.2008.12.026.

[34]

M. MeiC. Ou and X. Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790.  doi: 10.1137/090776342.

[35]

M. Mei and J. W. H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551-568.  doi: 10.1017/S0308210506000333.

[36]

M. MeiJ. W. H. SoM. Y. Li and S. S. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594.  doi: 10.1017/S0308210500003358.

[37]

Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.

[38]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715.

[39]

S. PanW. T. Li and G. Lin, Travelling wave fronts in nonlocal reaction–diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.  doi: 10.1007/s00033-007-7005-y.

[40]

K. Schumacher, Travelling-front solutions for integro-differential equations, I, J. Reine Angew. Math., 1980 (2009), 54-70.  doi: 10.1515/crll.1980.316.54.

[41]

H. L. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.

[42]

Y. J. SunW. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.  doi: 10.1016/j.jde.2011.04.020.

[43]

Y. J. SunL. ZhangW. T. Li and Z. C. Wang, Entire solutions in nonlocal monostable equations: Asymmetric case, Comm. Pure Appl. Anal., 18 (2019), 1049-1072.  doi: 10.3934/cpaa.2019051.

[44]

M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005.

[45]

Z. C. WangW. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1.

[46]

Z. C. WangW. T. Li and S. Ruan, Entire solutions in lattice delayed differential equations with nonlocal interaction: bistable cases, Math. Model. Nat. Phenom., 8 (2013), 78-103.  doi: 10.1051/mmnp/20138307.

[47]

Z. C. WangW. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.

[48]

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