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Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations
1. | School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, China |
2. | Department of Mathematics, National Central University, Chungli 32001, Taiwan |
In this work we consider the global asymptotic stability of pushed traveling fronts for one-dimensional monostable reaction-diffusion equations with monotone delayed reactions. Pushed traveling front is a special type of critical wave front which converges to zero more rapidly than the near non-critical wave fronts. Recently, Trofimchuk et al. [
References:
[1] |
O. Bonnefona, J. Garniera, F. Hamel and L. Roques,
Inside dynamics of delayed traveling waves, Math. Model. Nat. Phenom., 8 (2013), 42-59.
doi: 10.1051/mmnp/20138305. |
[2] |
X. Chen,
Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal
evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
[3] |
J. Garnier, T. Giletti, F. Hamel and L. Roques,
Inside dynamics of pulled and pushed fronts, J. Math. Pures Appl., 98 (2012), 428-449.
doi: 10.1016/j.matpur.2012.02.005. |
[4] |
S. A. Gourley and J. Wu, Delayed nonlocal diffusive systems in biological invasion and disease spread, in "Nonlinear Dynamics and Evolution Equations", Fields Inst. Commun. , Amer. Math. Soc. , Providence, RI, 48 (2006), 137-200. |
[5] |
X. Hou and Y. Li,
Local stability of traveling wave solutions of nonlinear reaction-diffusion
equations, Discrete Contin. Dyn. Syst., 15 (2006), 681-701.
doi: 10.3934/dcds.2006.15.681. |
[6] |
X. Liang and X.-Q. Zhao,
Asymptotic speeds of spread and traveling waves for monotone
semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.
doi: 10.1002/cpa.20154. |
[7] |
C. K. Lin and M. Mei,
On travelling wavefronts of the Nicholson's blowflies equation with
diffusion, Proc. Roy. Soc. Edinburgh(A), 140 (2010), 135-152.
doi: 10.1017/S0308210508000784. |
[8] |
S. Ma and X.-Q. Zhao,
Global asymptotic stability of minimal fronts in monostable lattice
equations, Discrete Contin. Dyn. Syst., 21 (2008), 259-275.
doi: 10.3934/dcds.2008.21.259. |
[9] |
M. Mei, C. K. Lin, C. T. Lin and J. W. H. So,
Traveling wavefronts for time-delayed reactiondiffusion equation: Ⅰ local nonlinearity, J. Differential Equations, 247 (2009), 495-510.
doi: 10.1016/j.jde.2008.12.026. |
[10] |
M. Mei, C. K. Lin, C. T. Lin and J. W. H. So,
Traveling wavefronts for time-delayed reactiondiffusion equation: Ⅱ nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529.
doi: 10.1016/j.jde.2008.12.020. |
[11] |
J. D. Murray,
Mathematical Biology Springer, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[12] |
K. W. Schaaf,
Asymptotic behavior and traveling wave solutions for parabolic functional
differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.
doi: 10.2307/2000859. |
[13] |
H. L. Smith and X.-Q. Zhao,
Global asymptotic stability of traveling waves in delayed reactiondiffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.
doi: 10.1137/S0036141098346785. |
[14] |
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. |
[15] |
A. Solar and S. Trofimchuk,
Speed selection and stability of wavefronts for delayed monostable
reaction-diffusion equations, J. Dynam. Differential Equations, 28 (2016), 1265-1292.
doi: 10.1007/s10884-015-9482-6. |
[16] |
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. |
[17] |
Z. C. Wang, W. T. Li and S. Ruan,
Traveling wave fronts in reaction-diffusion systems with
spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.
doi: 10.1016/j.jde.2005.08.010. |
[18] |
Z. C. Wang, W. T. Li and S. Ruan,
Existence and stability of traveling wave fronts in reaction
advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.
doi: 10.1016/j.jde.2007.03.025. |
[19] |
Z. C. Wang, W. T. Li and S. Ruan,
Travelling fronts in monostable equations with nonlocal
delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.
doi: 10.1007/s10884-008-9103-8. |
[20] |
J. Wu,
Theory and Applications of Partial Functional Differential Equations Springer, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[21] |
Y. Wu and X. Xing,
Stability of traveling waves with critical speeds for p-degree Fisher-type
equations, Discrete Contin. Dyn. Syst, 20 (2008), 1123-1139.
doi: 10.3934/dcds.2008.20.1123. |
[22] |
J. Wu and X. Zou,
Travelling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.
doi: 10.1023/A:1016690424892. |
[23] |
S. L. Wu, W. T. Li and S. Y. Liu,
Exponential stability of traveling fronts in monostable
reaction-advection-diffusion equations with non-local delay, Discrete Cont. Dyn. Syst., Ser. B, 17 (2012), 347-366.
doi: 10.3934/dcdsb.2012.17.347. |
show all references
References:
[1] |
O. Bonnefona, J. Garniera, F. Hamel and L. Roques,
Inside dynamics of delayed traveling waves, Math. Model. Nat. Phenom., 8 (2013), 42-59.
doi: 10.1051/mmnp/20138305. |
[2] |
X. Chen,
Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal
evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
[3] |
J. Garnier, T. Giletti, F. Hamel and L. Roques,
Inside dynamics of pulled and pushed fronts, J. Math. Pures Appl., 98 (2012), 428-449.
doi: 10.1016/j.matpur.2012.02.005. |
[4] |
S. A. Gourley and J. Wu, Delayed nonlocal diffusive systems in biological invasion and disease spread, in "Nonlinear Dynamics and Evolution Equations", Fields Inst. Commun. , Amer. Math. Soc. , Providence, RI, 48 (2006), 137-200. |
[5] |
X. Hou and Y. Li,
Local stability of traveling wave solutions of nonlinear reaction-diffusion
equations, Discrete Contin. Dyn. Syst., 15 (2006), 681-701.
doi: 10.3934/dcds.2006.15.681. |
[6] |
X. Liang and X.-Q. Zhao,
Asymptotic speeds of spread and traveling waves for monotone
semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.
doi: 10.1002/cpa.20154. |
[7] |
C. K. Lin and M. Mei,
On travelling wavefronts of the Nicholson's blowflies equation with
diffusion, Proc. Roy. Soc. Edinburgh(A), 140 (2010), 135-152.
doi: 10.1017/S0308210508000784. |
[8] |
S. Ma and X.-Q. Zhao,
Global asymptotic stability of minimal fronts in monostable lattice
equations, Discrete Contin. Dyn. Syst., 21 (2008), 259-275.
doi: 10.3934/dcds.2008.21.259. |
[9] |
M. Mei, C. K. Lin, C. T. Lin and J. W. H. So,
Traveling wavefronts for time-delayed reactiondiffusion equation: Ⅰ local nonlinearity, J. Differential Equations, 247 (2009), 495-510.
doi: 10.1016/j.jde.2008.12.026. |
[10] |
M. Mei, C. K. Lin, C. T. Lin and J. W. H. So,
Traveling wavefronts for time-delayed reactiondiffusion equation: Ⅱ nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529.
doi: 10.1016/j.jde.2008.12.020. |
[11] |
J. D. Murray,
Mathematical Biology Springer, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[12] |
K. W. Schaaf,
Asymptotic behavior and traveling wave solutions for parabolic functional
differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.
doi: 10.2307/2000859. |
[13] |
H. L. Smith and X.-Q. Zhao,
Global asymptotic stability of traveling waves in delayed reactiondiffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.
doi: 10.1137/S0036141098346785. |
[14] |
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. |
[15] |
A. Solar and S. Trofimchuk,
Speed selection and stability of wavefronts for delayed monostable
reaction-diffusion equations, J. Dynam. Differential Equations, 28 (2016), 1265-1292.
doi: 10.1007/s10884-015-9482-6. |
[16] |
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. |
[17] |
Z. C. Wang, W. T. Li and S. Ruan,
Traveling wave fronts in reaction-diffusion systems with
spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.
doi: 10.1016/j.jde.2005.08.010. |
[18] |
Z. C. Wang, W. T. Li and S. Ruan,
Existence and stability of traveling wave fronts in reaction
advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.
doi: 10.1016/j.jde.2007.03.025. |
[19] |
Z. C. Wang, W. T. Li and S. Ruan,
Travelling fronts in monostable equations with nonlocal
delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.
doi: 10.1007/s10884-008-9103-8. |
[20] |
J. Wu,
Theory and Applications of Partial Functional Differential Equations Springer, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[21] |
Y. Wu and X. Xing,
Stability of traveling waves with critical speeds for p-degree Fisher-type
equations, Discrete Contin. Dyn. Syst, 20 (2008), 1123-1139.
doi: 10.3934/dcds.2008.20.1123. |
[22] |
J. Wu and X. Zou,
Travelling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.
doi: 10.1023/A:1016690424892. |
[23] |
S. L. Wu, W. T. Li and S. Y. Liu,
Exponential stability of traveling fronts in monostable
reaction-advection-diffusion equations with non-local delay, Discrete Cont. Dyn. Syst., Ser. B, 17 (2012), 347-366.
doi: 10.3934/dcdsb.2012.17.347. |
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