Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author: Zhixian Yu

    * Corresponding author: Zhixian Yu 
Abstract Full Text(HTML) Related Papers Cited by
  • 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 $.

    Mathematics Subject Classification: Primary: 34K05; Secondary: 34A34, 34E05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
  • 加载中

Article Metrics

HTML views(312) PDF downloads(232) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint