American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020314

Novel entire solutions in a nonlocal 2-D discrete periodic media for bistable dynamics

Zhixian Yu 1,, , Rong Yuan 2, and  Shaohua Gan 3,

1. 

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2. 

The school of Mathematical Science, Beijing Normal University, Beijing 100875, China
3. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

* Corresponding author: Zhixian Yu

Received  June 2020 Revised  August 2020 Published  October 2020

This paper is concerned with novel entire solutions originating from three pulsating traveling fronts for nonlocal discrete periodic system (NDPS) on 2-D Lattices
$ \begin{align*} \label{eq1.1} u_{i,j}'(t) = \sum\limits_{k_1\in\mathbb{Z}\backslash \{0\}}\sum\limits_{k_2\in\mathbb{Z}\backslash \{0\} }J(k_1,k_2)\Big[u_{i-k_1,j-k_2}(t)- u_{i,j}(t)\Big]+ f_{i,j}(u_{i,j}(t)).\quad \end{align*} $
More precisely, let
$ \varphi_{i,j;k}(i cos\theta +j sin\theta+v_{k}t)\,\,(k = 1,2,3) $
 be the pulsating traveling front of NDPS with the wave speed
$ v_k $
 and connecting two different constant states, then NDPS admits an entire solution
$ u_{i,j}(t) $
, which satisfies
$ \begin{align*} &\ \lim\limits_{t\rightarrow-\infty}\Big\{ \sum\limits_{1\leq k\leq3}\sup\limits_{ p_{k-1}(t)\leq \xi\leq p_k(t)} |u_{i,j}(t)-\varphi_{i,j;k}(\xi+v_{k}t+\theta_{k})|\Big\} = 0, \end{align*} $
where
$ \xi = :i \cos\theta +j \sin\theta $
,
$ v_1<v_2<v_3 $
 and
$ \theta_{k}\,(k = 1,2) $
 is some constant,
$ p_0 = -\infty $
,
$ p_k(t): = -(v_k+v_{k+1})t/2\,\,(k = 1,2) $
 and
$ p_3 = +\infty $
.
Keywords: Entire solution, pulsating traveling front, discrete periodic media, super-sub solution.
Mathematics Subject Classification: Primary: 34K05; Secondary: 34A34, 34E05.
Citation: Zhixian Yu, Rong Yuan, Shaohua Gan. Novel entire solutions in a nonlocal 2-D discrete periodic media for bistable dynamics. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020314
References:
[1]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.  Google Scholar

[2]

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

[3]

Y.-Y. Chen, Entire solution originating from three fronts for a discrete diffusive equation, Tamkang J. Math., 48 (2017), 215-226.  doi: 10.5556/j.tkjm.48.2017.2442.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

X. Chen, J.-S. Guo and C.-C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal., 189 (2008), 189-236.  doi: 10.1007/s00205-007-0103-3.  Google Scholar

[7]

Y.-Y. Chen, J.-S. Guo, H. Ninomiya and C.-H. Yao, Entire solutions originating from monotones fronts to the Allen-Cahn equation, Physica D, 378-379 (2018), 1-19. doi: 10.1016/j.physd.2018.04.003.  Google Scholar

[8]

C.-P. Cheng, W.-T. Li and G. Lin, Travelling wave solutions in periodic monostable equations on a two-dimensional spatial lattice, IMA J. Appl. Math., 80 (2015), 1254-1272.  doi: 10.1093/imamat/hxu038.  Google Scholar

[9]

C.-P. Cheng, W.-T. Li and Z.-C. Wang, Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice, Nonlinear Anal. RWA, 13 (2012), 1873-1890.  doi: 10.1016/j.nonrwa.2011.12.016.  Google Scholar

[10]

C.-P. Cheng, W.-T. Li and Z.-C. Wang, Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 559-575.  doi: 10.3934/dcdsb.2010.13.559.  Google Scholar

[11]

C.-P. Cheng, W.-T. Li and Z.-C. Wang, Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice, IMA J. Appl. Math., 73 (2008), 592-618.  doi: 10.1093/imamat/hxn003.  Google Scholar

[12]

C.-P. Cheng, Y.-H. Su and Z. Feng, Wave propagation for monostable 2-D lattice differential equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350077, 11 pp. doi: 10.1142/S0218127413500776.  Google Scholar

[13]

F.-D. Dong, W.-T. Li and L. Zhang, Entire solutions in a two-dimensional nonlocal lattice dynamical system, Comm. Pure Appl. Anal., 17 (2018), 2517-2545.  doi: 10.3934/cpaa.2018120.  Google Scholar

[14]

P. C. Fife, Long time behavior of solutions of bistable diffusion equations, Arch. Ration. Mech. Anal., 70 (1979), 31-46.  doi: 10.1007/BF00276380.  Google Scholar

[15]

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.  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]

J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373.  Google Scholar

[18]

J.-S. Guo and C.-H. Wu, Front propagation for a two-dimensional periodic monostable lattice dynamical system, Discrete Contin. Dyn. Syst., 26 (2010), 197-223.  doi: 10.3934/dcds.2010.26.197.  Google Scholar

[19]

J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.  Google Scholar

[20]

J.-S. Guo and C.-H. Wu, Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system, Osaka J. Math., 45 (2008), 327-346.   Google Scholar

[21]

J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku Math. J., 62 (2010), 17-28.  doi: 10.2748/tmj/1270041024.  Google Scholar

[22]

S. Ma, P. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Anal., 65 (2006), 1858-1890.  doi: 10.1016/j.na.2005.10.042.  Google Scholar

[23]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014.  Google Scholar

[24]

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

[25]

Z.-C. Wang, W.-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.  Google Scholar

[26]

C.-C. Wu, Uniqueness of traveling waves for a two-dimensional bistable periodic lattice dynamical system, Abstr. Appl. Anal., 2012, Article ID 289168, 10 pages. doi: 10.1155/2012/289168.  Google Scholar

[27]

C.-H. Wu, A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations, 28 (2016), 317-338.  doi: 10.1007/s10884-016-9524-8.  Google Scholar

[28]

S.-L. Wu, G.-S. Chen and C.-H. Hsu, Entire solutions originating from multiple fronts of an epidemic model with nonlocal dispersal and bistable nonlinearity, J. Differential Equations, 265 (2018), 5520-5574.  doi: 10.1016/j.jde.2018.06.012.  Google Scholar

[29]

S.-L. Wu, G.-S. Chen and C.-H. Hsu, Pulsating traveling waves and entire solutions of a periodic lattice dynamical system, submitted. Google Scholar

[30]

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.  Google Scholar

[31]

S.-L. Wu, Z.-X. Shi and F.-Y. Yang, Entire solutions in periodic lattice dynamical systems, J. Differential Equations, 255 (2013), 3505-3535.  doi: 10.1016/j.jde.2013.07.049.  Google Scholar

show all references

References:
[1]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.  Google Scholar

[2]

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

[3]

Y.-Y. Chen, Entire solution originating from three fronts for a discrete diffusive equation, Tamkang J. Math., 48 (2017), 215-226.  doi: 10.5556/j.tkjm.48.2017.2442.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

X. Chen, J.-S. Guo and C.-C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal., 189 (2008), 189-236.  doi: 10.1007/s00205-007-0103-3.  Google Scholar

[7]

Y.-Y. Chen, J.-S. Guo, H. Ninomiya and C.-H. Yao, Entire solutions originating from monotones fronts to the Allen-Cahn equation, Physica D, 378-379 (2018), 1-19. doi: 10.1016/j.physd.2018.04.003.  Google Scholar

[8]

C.-P. Cheng, W.-T. Li and G. Lin, Travelling wave solutions in periodic monostable equations on a two-dimensional spatial lattice, IMA J. Appl. Math., 80 (2015), 1254-1272.  doi: 10.1093/imamat/hxu038.  Google Scholar

[9]

C.-P. Cheng, W.-T. Li and Z.-C. Wang, Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice, Nonlinear Anal. RWA, 13 (2012), 1873-1890.  doi: 10.1016/j.nonrwa.2011.12.016.  Google Scholar

[10]

C.-P. Cheng, W.-T. Li and Z.-C. Wang, Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 559-575.  doi: 10.3934/dcdsb.2010.13.559.  Google Scholar

[11]

C.-P. Cheng, W.-T. Li and Z.-C. Wang, Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice, IMA J. Appl. Math., 73 (2008), 592-618.  doi: 10.1093/imamat/hxn003.  Google Scholar

[12]

C.-P. Cheng, Y.-H. Su and Z. Feng, Wave propagation for monostable 2-D lattice differential equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350077, 11 pp. doi: 10.1142/S0218127413500776.  Google Scholar

[13]

F.-D. Dong, W.-T. Li and L. Zhang, Entire solutions in a two-dimensional nonlocal lattice dynamical system, Comm. Pure Appl. Anal., 17 (2018), 2517-2545.  doi: 10.3934/cpaa.2018120.  Google Scholar

[14]

P. C. Fife, Long time behavior of solutions of bistable diffusion equations, Arch. Ration. Mech. Anal., 70 (1979), 31-46.  doi: 10.1007/BF00276380.  Google Scholar

[15]

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.  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]

J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373.  Google Scholar

[18]

J.-S. Guo and C.-H. Wu, Front propagation for a two-dimensional periodic monostable lattice dynamical system, Discrete Contin. Dyn. Syst., 26 (2010), 197-223.  doi: 10.3934/dcds.2010.26.197.  Google Scholar

[19]

J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.  Google Scholar

[20]

J.-S. Guo and C.-H. Wu, Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system, Osaka J. Math., 45 (2008), 327-346.   Google Scholar

[21]

J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku Math. J., 62 (2010), 17-28.  doi: 10.2748/tmj/1270041024.  Google Scholar

[22]

S. Ma, P. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Anal., 65 (2006), 1858-1890.  doi: 10.1016/j.na.2005.10.042.  Google Scholar

[23]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014.  Google Scholar

[24]

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

[25]

Z.-C. Wang, W.-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.  Google Scholar

[26]

C.-C. Wu, Uniqueness of traveling waves for a two-dimensional bistable periodic lattice dynamical system, Abstr. Appl. Anal., 2012, Article ID 289168, 10 pages. doi: 10.1155/2012/289168.  Google Scholar

[27]

C.-H. Wu, A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations, 28 (2016), 317-338.  doi: 10.1007/s10884-016-9524-8.  Google Scholar

[28]

S.-L. Wu, G.-S. Chen and C.-H. Hsu, Entire solutions originating from multiple fronts of an epidemic model with nonlocal dispersal and bistable nonlinearity, J. Differential Equations, 265 (2018), 5520-5574.  doi: 10.1016/j.jde.2018.06.012.  Google Scholar

[29]

S.-L. Wu, G.-S. Chen and C.-H. Hsu, Pulsating traveling waves and entire solutions of a periodic lattice dynamical system, submitted. Google Scholar

[30]

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.  Google Scholar

[31]

S.-L. Wu, Z.-X. Shi and F.-Y. Yang, Entire solutions in periodic lattice dynamical systems, J. Differential Equations, 255 (2013), 3505-3535.  doi: 10.1016/j.jde.2013.07.049.  Google Scholar

[1]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159
[2]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002
[3]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
[4]

Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021018
[5]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[6]

Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020371
[7]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[8]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[9]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266
[10]

Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172
[11]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[12]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[13]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[14]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[15]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[16]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016
[17]

Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020162
[18]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405
[19]

Yan'e Wang, Nana Tian, Hua Nie. Positive solution branches of two-species competition model in open advective environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021006
[20]

Bingyan Liu, Xiongbing Ye, Xianzhou Dong, Lei Ni. Branching improved Deep Q Networks for solving pursuit-evasion strategy solution of spacecraft. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021016

2019 Impact Factor: 1.27

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences