# American Institute of Mathematical Sciences

• Previous Article
On a matrix-valued PDE characterizing a contraction metric for a periodic orbit
• DCDS-B Home
• This Issue
• Next Article
Propagation phenomena for a criss-cross infection model with non-diffusive susceptible population in periodic media
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 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, 2021, 26 (9) : 4815-4838. doi: 10.3934/dcdsb.2020314
##### References:

show all references

##### References:
 [1] Léo Girardin. Competition in periodic media:Ⅰ-Existence of pulsating fronts. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1341-1360. doi: 10.3934/dcdsb.2017065 [2] Shi-Liang Wu, Cheng-Hsiung Hsu. Propagation of monostable traveling fronts in discrete periodic media with delay. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2987-3022. doi: 10.3934/dcds.2018128 [3] 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 [4] Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002 [5] Yana Nec, Vladimir A Volpert, Alexander A Nepomnyashchy. Front propagation problems with sub-diffusion. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 827-846. doi: 10.3934/dcds.2010.27.827 [6] Fioralba Cakoni, Houssem Haddar. A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Problems & Imaging, 2007, 1 (3) : 443-456. doi: 10.3934/ipi.2007.1.443 [7] Bendong Lou. Periodic traveling waves of a mean curvature flow in heterogeneous media. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231 [8] Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993 [9] V.N. Malozemov, A.V. Omelchenko. On a discrete optimal control problem with an explicit solution. Journal of Industrial & Management Optimization, 2006, 2 (1) : 55-62. doi: 10.3934/jimo.2006.2.55 [10] Tran Ngoc Thach, Nguyen Huy Tuan, Donal O'Regan. Regularized solution for a biharmonic equation with discrete data. Evolution Equations & Control Theory, 2020, 9 (2) : 341-358. doi: 10.3934/eect.2020008 [11] Giovanni Scilla. Motion of discrete interfaces in low-contrast periodic media. Networks & Heterogeneous Media, 2014, 9 (1) : 169-189. doi: 10.3934/nhm.2014.9.169 [12] Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823 [13] 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 [14] Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101 [15] Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121 [16] 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 [17] Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103 [18] Adriana Navarro-Ramos, William Olvera-Lopez. A solution for discrete cost sharing problems with non rival consumption. Journal of Dynamics & Games, 2018, 5 (1) : 31-39. doi: 10.3934/jdg.2018004 [19] Irina Astashova, Josef Diblík, Evgeniya Korobko. Existence of a solution of discrete Emden-Fowler equation caused by continuous equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4159-4178. doi: 10.3934/dcdss.2021133 [20] Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure & Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487

2020 Impact Factor: 1.327