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Global stability of traveling waves for a spatially discrete diffusion system with time delay
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China |
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 $.
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] |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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] |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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