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June  2017, 10(3): 581-603. doi: 10.3934/dcdss.2017029

Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay

Zhao-Xing Yang 1, , Guo-Bao Zhang 1,, , Ge Tian 1, and  Zhaosheng Feng 2,

1. 

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China
2. 

School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA

* Corresponding author: Guo-Bao Zhang

Received  January 2016 Revised  December 2016 Published  February 2017

This paper is concerned with traveling waves for temporally delayed, spatially discrete reaction-diffusion equations without quasi-monotonicity. We first establish the existence of non-critical traveling waves (waves with speeds c>c*, where c* is minimal speed). Then by using the weighted energy method with a suitably selected weight function, we prove that all noncritical traveling waves Φ(x+ct) (monotone or nonmonotone) are time-asymptotically stable, when the initial perturbations around the wavefronts in a certain weighted Sobolev space are small.

Keywords: Spatially discrete reaction-diffusion equations, non-monotone traveling waves, stability, weighted energy.
Mathematics Subject Classification: Primary: 35K57, 34K20; Secondary: 92D25.
Citation: Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029
References:
[1]

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.  doi: 10.1006/jdeq.2001.4153.

[2]

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.

[3]

I.-Liang. Chern, M. Mei, X.-F. Yang and Q.-F. Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay, J. Differential Equations, 259 (2015), 1503-1541.  doi: 10.1016/j.jde.2015.03.003.

[4]

J. Fang, J. Wei and X.-Q. Zhao, Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. Lond. Ser. A., 466 (2010), 1919-1934.  doi: 10.1098/rspa.2009.0577.

[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.  doi: 10.3934/dcds.2005.12.193.

[6]

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.

[7]

S. J. Guo and J. Zimmer, Travelling wavefronts in discrete reaction-diffusion equations with nonlocal delay effects, arXiv: 1406.5321v1. doi: 10.1088/0951-7715/28/2/463.

[8]

S. J. Guo and J. Zimmer, Stability of traveling wavefronts in discrete reaction-diffusion equations with nonlocal delay effects, Nonlinearity, 28 (2015), 463-492.  doi: 10.1088/0951-7715/28/2/463.

[9]

S. A. Gourley, Linear stability of traveling fronts in an age-structured reaction-diffusion population model, Quart. J. Mech. Appl. Math., 58 (2005), 257-268.  doi: 10.1093/qjmamj/hbi012.

[10]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for non-monotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.

[11]

R. Huang, M. Mei, K.-J. Zhang and Q.-F. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations, Discrete Contin. Dyn. Syst., 36 (2016), 1331-1353.  doi: 10.3934/dcds.2016.36.1331.

[12]

C.-B. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations, 259 (2015), 1967-1989.  doi: 10.1016/j.jde.2015.03.025.

[13]

G. Lv and M.-X. Wang, Nonlinear stability of travelling wave fronts for delayed reaction diffusion equations, Nonlinearity, 23 (2010), 845-873.  doi: 10.1088/0951-7715/23/4/005.

[14]

C.-K. Lin, C.-T. Lin, Y. P. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084.  doi: 10.1137/120904391.

[15]

M. Mei, J. W.-H. So, M. Y. Li and S. S. P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594.  doi: 10.1017/S0308210500003358.

[16]

M. Mei and J. W.-H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551-568.  doi: 10.1017/S0308210506000333.

[17]

S. W. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277.  doi: 10.1016/j.jde.2007.03.014.

[18]

S. W. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  doi: 10.1016/j.jde.2005.05.004.

[19]

A. Solar and S. Trofimchuk, Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction, Nonlinearity, 28 (2015), 2027-2052.  doi: 10.1088/0951-7715/28/7/2027.

[20]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction, Discrete Contin. Dyn. Syst., 33 (2013), 2169-2187.  doi: 10.3934/dcds.2013.33.2169.

[21]

E. Trofimchuk, P. Alvarado and S. Trofimchuk, On the geometry of wave solutions of a delayed reaction-diffusion equation, J. Differential Equations, 246 (2009), 1422-1444.  doi: 10.1016/j.jde.2008.10.023.

[22]

S.-L. Wu, W.-T. Li and S.-Y. Liu, Asymptotic stability of traveling wave fronts in nonlocal reaction-diffusion equations with delay, J. Math. Anal. Appl., 360 (2009), 439-458.  doi: 10.1016/j.jmaa.2009.06.061.

[23]

S.-L. Wu, H.-Q. Zhao and S.-Y. Liu, Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability, Z. Angew. Math. Phys., 62 (2011), 377-397.  doi: 10.1007/s00033-010-0112-1.

[24]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651–687, J. Dynam. Differential Equations, 20 (2008), 531–533 (Erratum). doi: 10.1023/A:1016690424892.

[25]

G.-B. Zhang and R. Ma, Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution type crossing-monostable nonlinearity, Z. Angew. Math. Phys., 65 (2014), 819-844.  doi: 10.1007/s00033-013-0353-x.

show all references

References:
[1]

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.  doi: 10.1006/jdeq.2001.4153.

[2]

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.

[3]

I.-Liang. Chern, M. Mei, X.-F. Yang and Q.-F. Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay, J. Differential Equations, 259 (2015), 1503-1541.  doi: 10.1016/j.jde.2015.03.003.

[4]

J. Fang, J. Wei and X.-Q. Zhao, Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. Lond. Ser. A., 466 (2010), 1919-1934.  doi: 10.1098/rspa.2009.0577.

[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.  doi: 10.3934/dcds.2005.12.193.

[6]

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.

[7]

S. J. Guo and J. Zimmer, Travelling wavefronts in discrete reaction-diffusion equations with nonlocal delay effects, arXiv: 1406.5321v1. doi: 10.1088/0951-7715/28/2/463.

[8]

S. J. Guo and J. Zimmer, Stability of traveling wavefronts in discrete reaction-diffusion equations with nonlocal delay effects, Nonlinearity, 28 (2015), 463-492.  doi: 10.1088/0951-7715/28/2/463.

[9]

S. A. Gourley, Linear stability of traveling fronts in an age-structured reaction-diffusion population model, Quart. J. Mech. Appl. Math., 58 (2005), 257-268.  doi: 10.1093/qjmamj/hbi012.

[10]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for non-monotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.

[11]

R. Huang, M. Mei, K.-J. Zhang and Q.-F. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations, Discrete Contin. Dyn. Syst., 36 (2016), 1331-1353.  doi: 10.3934/dcds.2016.36.1331.

[12]

C.-B. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations, 259 (2015), 1967-1989.  doi: 10.1016/j.jde.2015.03.025.

[13]

G. Lv and M.-X. Wang, Nonlinear stability of travelling wave fronts for delayed reaction diffusion equations, Nonlinearity, 23 (2010), 845-873.  doi: 10.1088/0951-7715/23/4/005.

[14]

C.-K. Lin, C.-T. Lin, Y. P. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084.  doi: 10.1137/120904391.

[15]

M. Mei, J. W.-H. So, M. Y. Li and S. S. P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594.  doi: 10.1017/S0308210500003358.

[16]

M. Mei and J. W.-H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551-568.  doi: 10.1017/S0308210506000333.

[17]

S. W. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277.  doi: 10.1016/j.jde.2007.03.014.

[18]

S. W. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  doi: 10.1016/j.jde.2005.05.004.

[19]

A. Solar and S. Trofimchuk, Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction, Nonlinearity, 28 (2015), 2027-2052.  doi: 10.1088/0951-7715/28/7/2027.

[20]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction, Discrete Contin. Dyn. Syst., 33 (2013), 2169-2187.  doi: 10.3934/dcds.2013.33.2169.

[21]

E. Trofimchuk, P. Alvarado and S. Trofimchuk, On the geometry of wave solutions of a delayed reaction-diffusion equation, J. Differential Equations, 246 (2009), 1422-1444.  doi: 10.1016/j.jde.2008.10.023.

[22]

S.-L. Wu, W.-T. Li and S.-Y. Liu, Asymptotic stability of traveling wave fronts in nonlocal reaction-diffusion equations with delay, J. Math. Anal. Appl., 360 (2009), 439-458.  doi: 10.1016/j.jmaa.2009.06.061.

[23]

S.-L. Wu, H.-Q. Zhao and S.-Y. Liu, Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability, Z. Angew. Math. Phys., 62 (2011), 377-397.  doi: 10.1007/s00033-010-0112-1.

[24]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651–687, J. Dynam. Differential Equations, 20 (2008), 531–533 (Erratum). doi: 10.1023/A:1016690424892.

[25]

G.-B. Zhang and R. Ma, Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution type crossing-monostable nonlinearity, Z. Angew. Math. Phys., 65 (2014), 819-844.  doi: 10.1007/s00033-013-0353-x.

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