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September  2021, 26(9): 4815-4838. doi: 10.3934/dcdsb.2020314

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

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  September 2021 Early access  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 $
.
Citation: Zhixian Yu, Rong Yuan, Shaohua Gan. Novel entire solutions in a nonlocal 2-D discrete periodic media for bistable dynamics. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4815-4838. doi: 10.3934/dcdsb.2020314
References:
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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.

[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.

[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.

[4]

X. ChenS.-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.

[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.

[6]

X. ChenJ.-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.

[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.

[8]

C.-P. ChengW.-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.

[9]

C.-P. ChengW.-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.

[10]

C.-P. ChengW.-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.

[11]

C.-P. ChengW.-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.

[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.

[13]

F.-D. DongW.-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.

[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.

[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.

[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]

J.-S. GuoY. WangC.-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.

[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.

[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.

[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. 

[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.

[22]

S. MaP. 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.

[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.

[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.

[25]

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.

[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.

[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.

[28]

S.-L. WuG.-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.

[29]

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

[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.

[31]

S.-L. WuZ.-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.

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.

[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.

[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.

[4]

X. ChenS.-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.

[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.

[6]

X. ChenJ.-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.

[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.

[8]

C.-P. ChengW.-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.

[9]

C.-P. ChengW.-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.

[10]

C.-P. ChengW.-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.

[11]

C.-P. ChengW.-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.

[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.

[13]

F.-D. DongW.-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.

[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.

[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.

[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]

J.-S. GuoY. WangC.-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.

[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.

[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.

[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. 

[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.

[22]

S. MaP. 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.

[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.

[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.

[25]

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.

[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.

[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.

[28]

S.-L. WuG.-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.

[29]

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

[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.

[31]

S.-L. WuZ.-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.

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