January  2012, 32(1): 101-124. doi: 10.3934/dcds.2012.32.101

Traveling wave solution for a lattice dynamical system with convolution type nonlinearity

1. 

Department of Mathematics, Tamkang University, 151, Ying-Chuan Road, Tamsui, Taipei County 25137

2. 

Department of Mathematics, National Taiwan Normal University, 88, S-4, Ting Chou Road, Taipei 11677, Taiwan

Received  August 2010 Revised  December 2010 Published  September 2011

We study traveling wave solutions for a lattice dynamical system with convolution type nonlinearity. We consider the monostable case and discuss the asymptotic behaviors, monotonicity and uniqueness of traveling wave. First, we characterize the asymptotic behavior of wave profile at both wave tails. Next, we prove that any wave profile is strictly decreasing. Finally, we prove the uniqueness (up to translation) of wave profile for each given admissible wave speed.
Citation: Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101
References:
[1]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037.

[2]

P. W. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002.

[3]

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.

[4]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2 (1997), 125-160.

[5]

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.

[6]

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.

[7]

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.

[8]

J. Coville and L. Dupaigne, Travelling fronts in integrodifferential equations, C. R. Math. Acad. Sci. Paris, 337 (2003), 25-30.

[9]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030.

[10]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721.

[11]

S.-C. Fu, J.-S. Guo and S.-Y. Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations, Nonlinear Analysis Series A, 48 (2002), 1137-1149. doi: 10.1016/S0362-546X(00)00242-X.

[12]

W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, World Sci. Publ., 1 (1995), 187-199.

[13]

G. Lv, Asymptotic behavior of traveling fronts and entire solutions for a nonlinear monostable equation, Nonlinear Analysis, 72 (2010), 3659-3668. doi: 10.1016/j.na.2009.12.047.

[14]

S. Ma and X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.

[15]

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.

[16]

S. Ma, P. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Analysis, 65 (2006), 1858-1890. doi: 10.1016/j.na.2005.10.042.

[17]

K. Schumacher, Travelling-front solutions for integro-differential equations, J. Reine Angew. Math., 316 (1980), 54-70. doi: 10.1515/crll.1980.316.54.

[18]

P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409.

[19]

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.

[20]

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

[21]

B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62.

show all references

References:
[1]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037.

[2]

P. W. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002.

[3]

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.

[4]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2 (1997), 125-160.

[5]

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.

[6]

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.

[7]

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.

[8]

J. Coville and L. Dupaigne, Travelling fronts in integrodifferential equations, C. R. Math. Acad. Sci. Paris, 337 (2003), 25-30.

[9]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030.

[10]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721.

[11]

S.-C. Fu, J.-S. Guo and S.-Y. Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations, Nonlinear Analysis Series A, 48 (2002), 1137-1149. doi: 10.1016/S0362-546X(00)00242-X.

[12]

W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, World Sci. Publ., 1 (1995), 187-199.

[13]

G. Lv, Asymptotic behavior of traveling fronts and entire solutions for a nonlinear monostable equation, Nonlinear Analysis, 72 (2010), 3659-3668. doi: 10.1016/j.na.2009.12.047.

[14]

S. Ma and X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.

[15]

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.

[16]

S. Ma, P. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Analysis, 65 (2006), 1858-1890. doi: 10.1016/j.na.2005.10.042.

[17]

K. Schumacher, Travelling-front solutions for integro-differential equations, J. Reine Angew. Math., 316 (1980), 54-70. doi: 10.1515/crll.1980.316.54.

[18]

P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409.

[19]

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.

[20]

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

[21]

B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62.

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