# American Institute of Mathematical Sciences

• Previous Article
Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures
• DCDS Home
• This Issue
• Next Article
Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity
October  2015, 35(10): 5107-5131. doi: 10.3934/dcds.2015.35.5107

## Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system

 1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China 2 College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

Received  May 2014 Revised  January 2015 Published  April 2015

A delayed lattice dynamical system with non-local diffusion and interaction is considered in this paper. The exact asymptotics of the wave profile at both wave tails is derived, and all the wave profiles are shown to be strictly increasing. Moreover, we prove that the wave profile with a given admissible speed is unique up to translation. These results generalize earlier monotonicity, asymptotics and uniqueness results in the literature.
Citation: Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107
##### References:
 [1] M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts: Diekmann-Kaper theory of a nonlinear convolution equation re-visited, Math. Ann., 354 (2012), 73-109. doi: 10.1007/s00208-011-0722-8. [2] P. W. Bates and A. Chmaj, A discrete convolution model for phase transition, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189. [3] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Math., 22 (1991), 1-37. doi: 10.1007/BF01244896. [4] 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. [5] 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. [6] 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. [7] J. Coville, On uniqueness and monotonicity of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485. doi: 10.1007/s10231-005-0163-7. [8] S.-N. Chow, J. Mallet-Paret and W. X. Shen, Travelling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478. [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. [10] O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal. TMA., 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9. [11] J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations, Proc. Amer. Math. Soc., 139 (2010), 1361-1373. doi: 10.1090/S0002-9939-2010-10540-3. [12] J. Fang, J. Wei and X.-Q. Zhao, Spreading speed and travelling waves for non-monotone time-delayed lattice equations, Proc. Roy. Math. Soc. Lond. A., 466 (2010), 1919-1934. doi: 10.1098/rspa.2009.0577. [13] J.-S. Guo and Y.-C. Lin, Traveling wave solution for a lattice dynamical system with convolution type nonlinearity, Discrete Contin. Dyn. Syst., 32 (2012), 101-124. doi: 10.3934/dcds.2012.32.101. [14] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with application, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. [15] S. W. Ma and X. F. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87. doi: 10.1016/j.jde.2005.05.004. [16] S. W. Ma, P. X. Weng and X. F. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Analysis, 65 (2006), 1858-1890. doi: 10.1016/j.na.2005.10.042. [17] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spred and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [18] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ, 1941. [19] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409. [20] J. H. Wu and X. F. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. Differential Equations, 135 (1997), 315-357. doi: 10.1006/jdeq.1996.3232. [21] Z. X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays, Proc. Amer. Math. Soc., 140 (2012), 3853-3859. doi: 10.1090/S0002-9939-2012-11225-0. [22] P. A. Zhang and W. T. Li, Moonotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage, Nonlinear Anal. TMA., 72 (2010), 2178-2189. doi: 10.1016/j.na.2009.10.016. [23] 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] M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts: Diekmann-Kaper theory of a nonlinear convolution equation re-visited, Math. Ann., 354 (2012), 73-109. doi: 10.1007/s00208-011-0722-8. [2] P. W. Bates and A. Chmaj, A discrete convolution model for phase transition, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189. [3] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Math., 22 (1991), 1-37. doi: 10.1007/BF01244896. [4] 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. [5] 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. [6] 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. [7] J. Coville, On uniqueness and monotonicity of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485. doi: 10.1007/s10231-005-0163-7. [8] S.-N. Chow, J. Mallet-Paret and W. X. Shen, Travelling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478. [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. [10] O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal. TMA., 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9. [11] J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations, Proc. Amer. Math. Soc., 139 (2010), 1361-1373. doi: 10.1090/S0002-9939-2010-10540-3. [12] J. Fang, J. Wei and X.-Q. Zhao, Spreading speed and travelling waves for non-monotone time-delayed lattice equations, Proc. Roy. Math. Soc. Lond. A., 466 (2010), 1919-1934. doi: 10.1098/rspa.2009.0577. [13] J.-S. Guo and Y.-C. Lin, Traveling wave solution for a lattice dynamical system with convolution type nonlinearity, Discrete Contin. Dyn. Syst., 32 (2012), 101-124. doi: 10.3934/dcds.2012.32.101. [14] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with application, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. [15] S. W. Ma and X. F. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87. doi: 10.1016/j.jde.2005.05.004. [16] S. W. Ma, P. X. Weng and X. F. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Analysis, 65 (2006), 1858-1890. doi: 10.1016/j.na.2005.10.042. [17] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spred and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [18] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ, 1941. [19] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409. [20] J. H. Wu and X. F. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. Differential Equations, 135 (1997), 315-357. doi: 10.1006/jdeq.1996.3232. [21] Z. X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays, Proc. Amer. Math. Soc., 140 (2012), 3853-3859. doi: 10.1090/S0002-9939-2012-11225-0. [22] P. A. Zhang and W. T. Li, Moonotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage, Nonlinear Anal. TMA., 72 (2010), 2178-2189. doi: 10.1016/j.na.2009.10.016. [23] 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.
 [1] Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126 [2] Chin-Chin Wu. Monotonicity and uniqueness of wave profiles for a three components lattice dynamical system. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2813-2827. doi: 10.3934/dcds.2017121 [3] Thierry Horsin, Mohamed Ali Jendoubi. Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (3) : 999-1025. doi: 10.3934/cpaa.2022007 [4] Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255 [5] Tao Wang. Global dynamics of a non-local delayed differential equation in the half plane. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2475-2492. doi: 10.3934/cpaa.2014.13.2475 [6] Carl-Friedrich Kreiner, Johannes Zimmer. Heteroclinic travelling waves for the lattice sine-Gordon equation with linear pair interaction. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 915-931. doi: 10.3934/dcds.2009.25.915 [7] Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029 [8] Stig-Olof Londen, Hana Petzeltová. Convergence of solutions of a non-local phase-field system. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 653-670. doi: 10.3934/dcdss.2011.4.653 [9] Alejandro B. Aceves, Luis A. Cisneros-Ake, Antonmaria A. Minzoni. Asymptotics for supersonic traveling waves in the Morse lattice. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 975-994. doi: 10.3934/dcdss.2011.4.975 [10] Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707 [11] Cui-Ping Cheng, Ruo-Fan An. Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction. Electronic Research Archive, 2021, 29 (5) : 3535-3550. doi: 10.3934/era.2021051 [12] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [13] Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935 [14] Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347 [15] Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115 [16] Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203 [17] Sheng-Chen Fu, Je-Chiang Tsai. Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4041-4069. doi: 10.3934/dcds.2013.33.4041 [18] Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775 [19] Jianhua Huang, Xingfu Zou. Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 925-936. doi: 10.3934/dcds.2003.9.925 [20] Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems and Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036

2020 Impact Factor: 1.392