Entire solution originating from three fronts for a discrete diffusive equation

Yan Yu Chen


In this paper, we study a discrete diffusive equation with a bistable nonlinearity. For this equation, there are three types of traveling fronts. By constructing some suitable pairs of super-sub-solutions, we show that there are only two types of entire solutions originating from three fronts of this equation. These results show us some new dynamics of this discrete diffusive equation.


discrete diffusive equation; traveling front; entire solution; super-sub-solutions

Full Text:



X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations,184(2002), 549--569.

X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326(2003), 123--146.

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.

Y.-Y. Chen, J.-S. Guo, H. Ninomiya and C.-H. Yao, Entire solutions with merging three fronts to the Allen-Cahn equation, mathscidoc:1609.03007

Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8(2004), 15--32.

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.

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.

J.-S. Guo and Y.-C. Lin, Entire solutions for a discrete diffusive equation with bistable convolution type nonlinearity, Osaka Journal of Mathematics, 50(2013), 607--629.

Y.-J. Lin Guo, Eintire solutions for a discrete diffusive equation, Jornal of Mathematical Analysis and Applications, 347(2008), 450--458.

F.Hamel and N.Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52(1999), 1255--1276.

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47(1987), 556--572.

Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Diff. Eq., 18(2006), 841--861.

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.

S.-L. Wu, Z.-X. Shih and F.-Y. Yang, Entire solutions in periodic lattice dynamical systems, J. Diff. Eq., 255(2013), 62--84.

H.Yagisita, Backward global solutions characterizing annihilation dynamics of

traveling fronts, Publ. Res. Inst. Math. Sci., 39(2003), 117--164.

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equations}, SIAM J. Math. Anal., 22(1991), 1016--1020.

B. Zinner, Existence of traveling wavefronts for the discrete Nagumo equations, J. Differential Equations, 96(1992), 1--27.

DOI: http://dx.doi.org/10.5556/j.tkjm.48.2017.2442

Sponsored by Tamkang University | ISSN 0049-2930 (Print), ISSN 2073-9826 (Online) | Powered by MathJax