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October  2018, 11(5): 1235-1253. doi: 10.3934/krm.2018048

Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations

Yicheng Jiang and  Kaijun Zhang ,

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

* Corresponding author

Received  June 2017 Revised  July 2017 Published  May 2018

Fund Project: The first author is supported by NSFC grant (No.11571066) and the second author is supported by NSFC grant (No.11771071).

This paper is concerned with the stability of noncritical/critical traveling waves for nonlocal time-delayed reaction-diffusion equation. When the birth rate function is non-monotone, the solution of the delayed equation is proved to converge time-exponentially to some (monotone or non-monotone) traveling wave profile with wave speed $c>c_*$, where $c_*>0$ is the minimum wave speed, when the initial data is a small perturbation around the wave. However, for the critical traveling waves ($c = c_*$), the time-asymptotical stability is only obtained, and the decay rate is not gotten due to some technical restrictions. The proof approach is based on the combination of the anti-weighted method and the nonlinear Halanay inequality but with some new development.

Keywords: Traveling wave, time delay, nonlocal reaction-diffusion equations, $L^2$-weighted energy, stability.
Mathematics Subject Classification: Primary: 35K57, 35C07; Secondary: 35K58, 92D25.
Citation: Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic & Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048
References:
[1]

M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts, Math. Ann., 354 (2012), 73-109.  doi: 10.1007/s00208-011-0722-8.  Google Scholar

[2]

I. L. 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.  Google Scholar

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T. Faria, W. Huang and J. Wu, Traveling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261.  doi: 10.1098/rspa.2005.1554.  Google Scholar

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T. Faria and S. Trofimchuk, Nonmonotone traveling waves in single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.  doi: 10.1016/j.jde.2006.05.006.  Google Scholar

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T. Faria and S. Trofimchuk, Positive heteroclinics and traveling waves for scalar population models with a single delay, Appl. Math. Comput., 185 (2007), 594-603.  doi: 10.1016/j.amc.2006.07.059.  Google Scholar

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A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. Lond. Math. Soc., 89 (2014), 47-68.  doi: 10.1112/jlms/jdt050.  Google Scholar

[8]

S. A. Gourley and J. Wu, Delayed nonlocal diffusive system in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200.   Google Scholar

[9]

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0.  Google Scholar

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R. Huang, M. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3621-3649.  doi: 10.3934/dcds.2012.32.3621.  Google Scholar

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

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Y. C. Jiang and K. J. Zhang, Time-delayed reaction-diffusion equation with boundary effect: (Ⅰ) converegence to non-critical traveling waves, Applicable Analysis, 97 (2018), 230-254.  doi: 10.1080/00036811.2016.1258696.  Google Scholar

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W. T. Li, S. G. Ruan and Z. C. Wang, On the diffusive Nicholson's blowflies equation with nonlocal delays, J. Nonlinear Sci., 17 (2007), 505-525.  doi: 10.1007/s00332-007-9003-9.  Google Scholar

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

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C. K. Lin and M. Mei, On traveling wavefronts of the Nicholson's blowflies equations with diffusion, Proc. Roy. Soc. Edinburgh Set. A, 140 (2010), 135-152.  doi: 10.1017/S0308210508000784.  Google Scholar

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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.  Google Scholar

[17]

A. Matsumura and M. Mei, Convergence to traveling fronts of solutions of the $p$-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal., 146 (1999), 1-22.  doi: 10.1007/s002050050134.  Google Scholar

[18]

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

[19]

M. Mei, C. K. Lin, C. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation, (Ⅱ) Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529.  doi: 10.1016/j.jde.2008.12.020.  Google Scholar

[20]

M. Mei, C. H. Ou and X. Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Appl. Math., 42 (2010), 2762–2790; erratum, SIAM J. Appl. Math., 44 (2012), 538–540. doi: 10.1137/110850633.  Google Scholar

[21]

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

[22]

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

[23]

M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Num. Anal. Model Ser. B, 2 (2011), 379-401.   Google Scholar

[24]

A. J. Nicholson, Competition for food amongst Lucilia Cuprina larvae, Proceedings of the 8th International Congress of Entomology, Stockhom, (1984), 227–281. Google Scholar

[25]

A. J. Nicholson, An outline of dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65.  doi: 10.1071/ZO9540009.  Google Scholar

[26]

J. W. H. So and Y. Yang, Dirichlet problem for the diffusion Nicholson's blowflies equation, J. Differential Equations, 150 (1998), 317-348.  doi: 10.1006/jdeq.1998.3489.  Google Scholar

[27]

J. So and X. Zou, Traveling waves for the diffusion Nicholson's blowflies equation, Appl. Math. Comput., 122 (2001), 385-392.  doi: 10.1016/S0096-3003(00)00055-2.  Google Scholar

[28]

E. Trofimchuk, V. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay, J. Differential Equations, 245 (2008), 2307-2332.  doi: 10.1016/j.jde.2008.06.023.  Google Scholar

[29]

E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion with delay, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 407-423.  doi: 10.3934/dcds.2008.20.407.  Google Scholar

[30]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems, J. Dyn. Differ. Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

show all references

References:
[1]

M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts, Math. Ann., 354 (2012), 73-109.  doi: 10.1007/s00208-011-0722-8.  Google Scholar

[2]

I. L. 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.  Google Scholar

[3]

J. Fang and X. Q. Zhao, Esistence and uniqueness of traveling waves for non-monotone integral equations with in applications, J. Differential Equations, 248 (2010), 2199-2226.  doi: 10.1016/j.jde.2010.01.009.  Google Scholar

[4]

T. Faria, W. Huang and J. Wu, Traveling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261.  doi: 10.1098/rspa.2005.1554.  Google Scholar

[5]

T. Faria and S. Trofimchuk, Nonmonotone traveling waves in single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.  doi: 10.1016/j.jde.2006.05.006.  Google Scholar

[6]

T. Faria and S. Trofimchuk, Positive heteroclinics and traveling waves for scalar population models with a single delay, Appl. Math. Comput., 185 (2007), 594-603.  doi: 10.1016/j.amc.2006.07.059.  Google Scholar

[7]

A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. Lond. Math. Soc., 89 (2014), 47-68.  doi: 10.1112/jlms/jdt050.  Google Scholar

[8]

S. A. Gourley and J. Wu, Delayed nonlocal diffusive system in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200.   Google Scholar

[9]

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0.  Google Scholar

[10]

R. Huang, M. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3621-3649.  doi: 10.3934/dcds.2012.32.3621.  Google Scholar

[11]

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

[12]

Y. C. Jiang and K. J. Zhang, Time-delayed reaction-diffusion equation with boundary effect: (Ⅰ) converegence to non-critical traveling waves, Applicable Analysis, 97 (2018), 230-254.  doi: 10.1080/00036811.2016.1258696.  Google Scholar

[13]

W. T. Li, S. G. Ruan and Z. C. Wang, On the diffusive Nicholson's blowflies equation with nonlocal delays, J. Nonlinear Sci., 17 (2007), 505-525.  doi: 10.1007/s00332-007-9003-9.  Google Scholar

[14]

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

[15]

C. K. Lin and M. Mei, On traveling wavefronts of the Nicholson's blowflies equations with diffusion, Proc. Roy. Soc. Edinburgh Set. A, 140 (2010), 135-152.  doi: 10.1017/S0308210508000784.  Google Scholar

[16]

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.  Google Scholar

[17]

A. Matsumura and M. Mei, Convergence to traveling fronts of solutions of the $p$-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal., 146 (1999), 1-22.  doi: 10.1007/s002050050134.  Google Scholar

[18]

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

[19]

M. Mei, C. K. Lin, C. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation, (Ⅱ) Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529.  doi: 10.1016/j.jde.2008.12.020.  Google Scholar

[20]

M. Mei, C. H. Ou and X. Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Appl. Math., 42 (2010), 2762–2790; erratum, SIAM J. Appl. Math., 44 (2012), 538–540. doi: 10.1137/110850633.  Google Scholar

[21]

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

[22]

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

[23]

M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Num. Anal. Model Ser. B, 2 (2011), 379-401.   Google Scholar

[24]

A. J. Nicholson, Competition for food amongst Lucilia Cuprina larvae, Proceedings of the 8th International Congress of Entomology, Stockhom, (1984), 227–281. Google Scholar

[25]

A. J. Nicholson, An outline of dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65.  doi: 10.1071/ZO9540009.  Google Scholar

[26]

J. W. H. So and Y. Yang, Dirichlet problem for the diffusion Nicholson's blowflies equation, J. Differential Equations, 150 (1998), 317-348.  doi: 10.1006/jdeq.1998.3489.  Google Scholar

[27]

J. So and X. Zou, Traveling waves for the diffusion Nicholson's blowflies equation, Appl. Math. Comput., 122 (2001), 385-392.  doi: 10.1016/S0096-3003(00)00055-2.  Google Scholar

[28]

E. Trofimchuk, V. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay, J. Differential Equations, 245 (2008), 2307-2332.  doi: 10.1016/j.jde.2008.06.023.  Google Scholar

[29]

E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion with delay, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 407-423.  doi: 10.3934/dcds.2008.20.407.  Google Scholar

[30]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems, J. Dyn. Differ. Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

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