# American Institute of Mathematical Sciences

April  2016, 36(4): 2329-2346. doi: 10.3934/dcds.2016.36.2329

## Entire solutions with merging fronts to a bistable periodic lattice dynamical system

 1 Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071 2 Department of Mathematics, National Central University, Chung-Li 32001

Received  September 2014 Revised  July 2015 Published  September 2015

We are interested in finding entire solutions of a bistable periodic lattice dynamical system. By constructing appropriate super- and subsolutions of the system, we establish two different types of merging-front entire solutions. The first type can be characterized by two monostable fronts merging and converging to a single bistable front; while the second type is a solution which behaves as a monostable front merging with a bistable front and one chases another from the same side of $x$-axis. For this discrete and spatially periodic system, we have to emphasize that there has no symmetry between the increasing and decreasing pulsating traveling fronts, which increases the difficulty of construction of the super- and subsolutions.
Citation: Shi-Liang Wu, Cheng-Hsiung Hsu. Entire solutions with merging fronts to a bistable periodic lattice dynamical system. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2329-2346. doi: 10.3934/dcds.2016.36.2329
##### References:
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##### References:
 [1] X. Chen, J.-S. Guo and C. C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Rational Mech. Anal., 189 (2008), 189-236. doi: 10.1007/s00205-007-0103-3. [2] S.-N. Chow, J. Mallet-Paret and W. Shen, Travelling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478. [3] P.-C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer Verlag, 1979. [4] 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. [5] 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. [6] J.-S. Guo and C. H. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations, 246 (2009), 3818-3833. doi: 10.1016/j.jde.2009.03.010. [7] Y.-J. L. Guo, Entire solutions for a discrete diffusive equation, J. Math. Anal. Appl., 347 (2008), 450-458. doi: 10.1016/j.jmaa.2008.03.076. [8] F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276. doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W. [9] W.-T. Li, N.-W. Liu and Z.-C. Wang, Entire solutions in reaction-advection-diffusion equations in cylinders, J. Math. Pures Appl., 90 (2008), 492-504. doi: 10.1016/j.matpur.2008.07.002. [10] W.-T. Li, Z.-C. Wang and J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129. doi: 10.1016/j.jde.2008.03.023. [11] X. Liang and X. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903. doi: 10.1016/j.jfa.2010.04.018. [12] N.-W. Liu, W.-T. Li and Z.-C. Wang, Entire solutions of reaction-advection-diffusion equations with bistable nonlinearity in cylinders, J. Differential Equations, 246 (2009), 4249-4267. doi: 10.1016/j.jde.2008.12.005. [13] 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. [14] S. Ma and X. Zhao, Global asymptotic stability of minimal fronts in monostable lattice equations, Discrete Contin. Dyn. Syst., 21 (2008), 259-275. doi: 10.3934/dcds.2008.21.259. [15] 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. [16] Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240. doi: 10.1137/080723715. [17] N. Shigesada and K. Kawasaki, Biological invasions: theory and practice, Oxford Series in Ecology and Evolution, Oxford, Oxford University Press, 1997. [18] Y.-J. Sun, W.-T. Li and Z.-C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581. doi: 10.1016/j.jde.2011.04.020. [19] Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084. doi: 10.1090/S0002-9947-08-04694-1. [20] Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420. doi: 10.1137/080727312. [21] Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in lattice delayed differential equations with nonlocal interaction: Bistable case, Math. Model. Nat. Phenom., 8 (2013), 78-103. doi: 10.1051/mmnp/20138307. [22] M.-X. Wang and G.-Y. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delay, Nonlinearity, 23 (2010), 1609-1630. doi: 10.1088/0951-7715/23/7/005. [23] 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. [24] S.-L. Wu, Y.-J. Sun and S.-Y. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity, Discrete Contin. Dyn. Syst., 33 (2013), 921-946. doi: 10.3934/dcds.2013.33.921. [25] S.-L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533. doi: 10.1007/s10884-013-9293-6.
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