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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020152
References:
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[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.  Google Scholar

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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.  Google Scholar

[2]

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[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.  Google Scholar

[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.  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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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