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Exponential stability of 1-d wave equation with the boundary time delay based on the interior control
Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay
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 |
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.
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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