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Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics
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 |
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. |
show all references
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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