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June  2017, 37(6): 3467-3486. doi: 10.3934/dcds.2017147

Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations

Shi-Liang Wu 1,, , Tong-Chang Niu 1, and  Cheng-Hsiung Hsu 2,

1. 

School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, China
2. 

Department of Mathematics, National Central University, Chungli 32001, Taiwan

Corresponding author

Received  April 2016 Revised  January 2017 Published  February 2017

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. [16] proved the existence and uniqueness of pushed traveling fronts of the considered equation when the reaction term lost the sub-tangency condition. In this article, using the comparison method via a pair of super-and sub-solution and squeezing technique, we prove that the pushed traveling fronts are globally exponentially stable. This also gives an affirmative answer to an open problem presented by Solar and Trofimchuk [14].

Keywords: Global asymptotic stability, pushed traveling front, sub-and supersolution method, squeezing technique.
Mathematics Subject Classification: Primary: 35K57; Secondary: 92D25, 35B40.
Citation: Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147
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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