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A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations
Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations
School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
A. Gomez and S. Trofimchuk,
Global continuation of monotone wavefronts, J. Lond. Math. Soc., 89 (2014), 47-68.
doi: 10.1112/jlms/jdt050. |
[8] |
S. A. Gourley and J. Wu,
Delayed nonlocal diffusive system in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[24] |
A. J. Nicholson, Competition for food amongst Lucilia Cuprina larvae, Proceedings of the 8th International Congress of Entomology, Stockhom, (1984), 227–281. |
[25] |
A. J. Nicholson,
An outline of dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65.
doi: 10.1071/ZO9540009. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
A. Gomez and S. Trofimchuk,
Global continuation of monotone wavefronts, J. Lond. Math. Soc., 89 (2014), 47-68.
doi: 10.1112/jlms/jdt050. |
[8] |
S. A. Gourley and J. Wu,
Delayed nonlocal diffusive system in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[24] |
A. J. Nicholson, Competition for food amongst Lucilia Cuprina larvae, Proceedings of the 8th International Congress of Entomology, Stockhom, (1984), 227–281. |
[25] |
A. J. Nicholson,
An outline of dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65.
doi: 10.1071/ZO9540009. |
[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. |
[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. |
[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. |
[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. |
[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. |
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