# American Institute of Mathematical Sciences

## 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  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 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 & 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. 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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. 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