American Institute of Mathematical Sciences

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doi: 10.3934/era.2021003

Global stability of traveling waves for a spatially discrete diffusion system with time delay

Ting Liu and  Guo-Bao Zhang ,

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

* Corresponding author: Guo-Bao Zhang

Received  September 2020 Revised  November 2020 Published  January 2021

Fund Project: The second author is supported by NSF of China (11861056)

This article deals with the global stability of traveling waves of a spatially discrete diffusion system with time delay and without quasi-monotonicity. Using the Fourier transform and the weighted energy method with a suitably selected weighted function, we prove that the monotone or non-monotone traveling waves are exponentially stable in $ L^\infty(\mathbb{R})\times L^\infty(\mathbb{R}) $ with the exponential convergence rate $ e^{-\mu t} $ for some constant $ \mu>0 $.

Keywords: Spatially discrete diffusion system, traveling waves, global stability, weighted energy method, the Fourier transform.
Mathematics Subject Classification: Primary: 35K57, 35B35; Secondary: 92D30.
Citation: Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, doi: 10.3934/era.2021003
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.  Google Scholar

[2]

G.-S. Chen, S.-L. Wu and C.-H. Hsu, Stability of traveling wavefronts for a discrete diffusive competition system with three species, J. Math. Anal. Appl., 474 (2019), 909-930.  doi: 10.1016/j.jmaa.2019.01.079.  Google Scholar

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I.-L. Chern, M. Mei, X. Yang and Q. 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.  Google Scholar

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S. 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.  Google Scholar

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C.-H. Hsu, J.-J. Lin and T.-S. Yang, Traveling wave solutions for delayed lattice reaction-diffusion systems, IMA J. Appl. Math., 80 (2015), 302-323.  doi: 10.1093/imamat/hxt039.  Google Scholar

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C.-H. Hsu, T.-S. Yang and Z. Yu, Existence and exponential stability of traveling waves for delayed reaction-diffusion systems, Nonlinearity, 31 (2018), 838-863.  doi: 10.1088/1361-6544/aa99a1.  Google Scholar

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C. 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.  Google Scholar

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Y. Li, W.-T. Li and Y.-R. Yang, Stability of traveling waves of a diffusive susceptible-infective-removed (SIR) epidemic model, J. Math. Phys., 57 (2016), 041504, 28 pp. doi: 10.1063/1.4947106.  Google Scholar

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C.-K. Lin, C.-T. Lin, Y. 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.  Google Scholar

[11]

Z. Ma, R. Yuan, Y. Wang and X. Wu, Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system, Commu. Pure Appl. Anal., 18 (2019), 2069-2091.  doi: 10.3934/cpaa.2019093.  Google Scholar

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M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (I) local nonlinearity, J. Differential Equations, 247 (2009), 495-510.  doi: 10.1016/j.jde.2008.12.026.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

M. Mei, K. Zhang and Q. Zhang, Global stability of traveling waves with oscillations for time-delayed reaction-diffusion equations, Int. J. Numer. Anal. Model., 16 (2019), 375-397.   Google Scholar

[16]

H. L. Smith and X.-Q. Zhao, Global asymptotical stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.  Google Scholar

[17]

T. Su and G.-B. Zhang, Global stability of non-monotone noncritical traveling waves for a discrete diffusion equation with a convolution type nonlinearity, Taiwanese J. Math., 24 (2020), 937-957.  doi: 10.11650/tjm/190901.  Google Scholar

[18]

S. Su and G.-B. Zhang, Global stability of traveling waves for delay reaction-diffusion systems without quasi-momotonicity, Electron. J. Differential Equations, (2020), Paper No. 46, 18 pp.  Google Scholar

[19]

G. Tian and G.-B. Zhang, Stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system, J. Math. Anal. Appl., 447 (2017), 222-242.  doi: 10.1016/j.jmaa.2016.10.012.  Google Scholar

[20]

G. Tian, G. Zhang and Z. Yang, Stability of nonmonotone critical traveling waves for spatially discrete reaction-diffusion equations with time delay, Turkish J. Math., 41 (2017), 655-680.  doi: 10.3906/mat-1601-19.  Google Scholar

[21]

S.-L. Wu and S.-Y. Liu, Existence and uniqueness of traveling waves for non-monotone integral equations with application, J. Math. Anal. Appl., 365 (2010), 729-741.  doi: 10.1016/j.jmaa.2009.11.028.  Google Scholar

[22]

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.  Google Scholar

[23]

T. Xu, S. Ji, R. Huang, M. Mei and J. Yin, Theoretical and numerical studies on global stability of traveling waves with oscillation for time-delayed nonlocal dispersion equations, Int. J. Numer. Anal. Model., 17 (2020), 68-86.   Google Scholar

[24]

Y.-R. Yang, W.-T. Li and S.-L. Wu, Stability of traveling waves in a monostable delayed system without quasi-monotonicity, Nonlinear Anal. Real World Appl., 14 (2013), 1511-1526.  doi: 10.1016/j.nonrwa.2012.10.015.  Google Scholar

[25]

Z. Yang and G. Zhang, Stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity, Sci. China Math., 61 (2018), 1789-1806.  doi: 10.1007/s11425-017-9175-2.  Google Scholar

[26]

Z.-X. Yang, G.-B. Zhang, G. Tian and Z. Feng, Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 581-603.  doi: 10.3934/dcdss.2017029.  Google Scholar

[27]

Z. Yu and C.-H. Hsu, Wave propagation and its stability for a class of discrete diffusion systems, Z. Angew. Math. Phys., 71 (2020), 194. doi: 10.1007/s00033-020-01423-4.  Google Scholar

[28]

G.-B. Zhang, Global stability of non-monotone traveling wave solutions for a nonlocal dispersal equation with time delay, J. Math. Anal. Appl., 475 (2019), 605-627.  doi: 10.1016/j.jmaa.2019.02.058.  Google Scholar

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.  Google Scholar

[2]

G.-S. Chen, S.-L. Wu and C.-H. Hsu, Stability of traveling wavefronts for a discrete diffusive competition system with three species, J. Math. Anal. Appl., 474 (2019), 909-930.  doi: 10.1016/j.jmaa.2019.01.079.  Google Scholar

[3]

I.-L. Chern, M. Mei, X. Yang and Q. 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.  Google Scholar

[4]

S. A. Gourley and Y. Kuang, Wavefronts and global stability in a time-delayed population model with stage structure, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 1563-1579.  doi: 10.1098/rspa.2002.1094.  Google Scholar

[5]

S. 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.  Google Scholar

[6]

C.-H. Hsu, J.-J. Lin and T.-S. Yang, Traveling wave solutions for delayed lattice reaction-diffusion systems, IMA J. Appl. Math., 80 (2015), 302-323.  doi: 10.1093/imamat/hxt039.  Google Scholar

[7]

C.-H. Hsu, T.-S. Yang and Z. Yu, Existence and exponential stability of traveling waves for delayed reaction-diffusion systems, Nonlinearity, 31 (2018), 838-863.  doi: 10.1088/1361-6544/aa99a1.  Google Scholar

[8]

C. 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.  Google Scholar

[9]

Y. Li, W.-T. Li and Y.-R. Yang, Stability of traveling waves of a diffusive susceptible-infective-removed (SIR) epidemic model, J. Math. Phys., 57 (2016), 041504, 28 pp. doi: 10.1063/1.4947106.  Google Scholar

[10]

C.-K. Lin, C.-T. Lin, Y. 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.  Google Scholar

[11]

Z. Ma, R. Yuan, Y. Wang and X. Wu, Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system, Commu. Pure Appl. Anal., 18 (2019), 2069-2091.  doi: 10.3934/cpaa.2019093.  Google Scholar

[12]

M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (I) local nonlinearity, J. Differential Equations, 247 (2009), 495-510.  doi: 10.1016/j.jde.2008.12.026.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

M. Mei, K. Zhang and Q. Zhang, Global stability of traveling waves with oscillations for time-delayed reaction-diffusion equations, Int. J. Numer. Anal. Model., 16 (2019), 375-397.   Google Scholar

[16]

H. L. Smith and X.-Q. Zhao, Global asymptotical stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.  Google Scholar

[17]

T. Su and G.-B. Zhang, Global stability of non-monotone noncritical traveling waves for a discrete diffusion equation with a convolution type nonlinearity, Taiwanese J. Math., 24 (2020), 937-957.  doi: 10.11650/tjm/190901.  Google Scholar

[18]

S. Su and G.-B. Zhang, Global stability of traveling waves for delay reaction-diffusion systems without quasi-momotonicity, Electron. J. Differential Equations, (2020), Paper No. 46, 18 pp.  Google Scholar

[19]

G. Tian and G.-B. Zhang, Stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system, J. Math. Anal. Appl., 447 (2017), 222-242.  doi: 10.1016/j.jmaa.2016.10.012.  Google Scholar

[20]

G. Tian, G. Zhang and Z. Yang, Stability of nonmonotone critical traveling waves for spatially discrete reaction-diffusion equations with time delay, Turkish J. Math., 41 (2017), 655-680.  doi: 10.3906/mat-1601-19.  Google Scholar

[21]

S.-L. Wu and S.-Y. Liu, Existence and uniqueness of traveling waves for non-monotone integral equations with application, J. Math. Anal. Appl., 365 (2010), 729-741.  doi: 10.1016/j.jmaa.2009.11.028.  Google Scholar

[22]

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.  Google Scholar

[23]

T. Xu, S. Ji, R. Huang, M. Mei and J. Yin, Theoretical and numerical studies on global stability of traveling waves with oscillation for time-delayed nonlocal dispersion equations, Int. J. Numer. Anal. Model., 17 (2020), 68-86.   Google Scholar

[24]

Y.-R. Yang, W.-T. Li and S.-L. Wu, Stability of traveling waves in a monostable delayed system without quasi-monotonicity, Nonlinear Anal. Real World Appl., 14 (2013), 1511-1526.  doi: 10.1016/j.nonrwa.2012.10.015.  Google Scholar

[25]

Z. Yang and G. Zhang, Stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity, Sci. China Math., 61 (2018), 1789-1806.  doi: 10.1007/s11425-017-9175-2.  Google Scholar

[26]

Z.-X. Yang, G.-B. Zhang, G. Tian and Z. Feng, Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 581-603.  doi: 10.3934/dcdss.2017029.  Google Scholar

[27]

Z. Yu and C.-H. Hsu, Wave propagation and its stability for a class of discrete diffusion systems, Z. Angew. Math. Phys., 71 (2020), 194. doi: 10.1007/s00033-020-01423-4.  Google Scholar

[28]

G.-B. Zhang, Global stability of non-monotone traveling wave solutions for a nonlocal dispersal equation with time delay, J. Math. Anal. Appl., 475 (2019), 605-627.  doi: 10.1016/j.jmaa.2019.02.058.  Google Scholar

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