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

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

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].

Citation: Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147
References:
[1]

O. BonnefonaJ. GarnieraF. 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. GarnierT. GilettiF. 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. MeiC. K. LinC. 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. MeiC. K. LinC. 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. TrofimchukM. 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. WangW. 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. WangW. 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. WangW. 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. WuW. 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. BonnefonaJ. GarnieraF. 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. GarnierT. GilettiF. 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. MeiC. K. LinC. 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. MeiC. K. LinC. 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. TrofimchukM. 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. WangW. 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. WangW. 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. WangW. 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. WuW. 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.

[1]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101

[2]

Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111

[3]

Yana Nec, Vladimir A Volpert, Alexander A Nepomnyashchy. Front propagation problems with sub-diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 827-846. doi: 10.3934/dcds.2010.27.827

[4]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. Pushed traveling fronts in monostable equations with monotone delayed reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2169-2187. doi: 10.3934/dcds.2013.33.2169

[5]

Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885

[6]

Hongmei Cheng, Rong Yuan. Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 3007-3022. doi: 10.3934/dcdsb.2017160

[7]

Christian Lax, Sebastian Walcher. A note on global asymptotic stability of nonautonomous master equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2143-2149. doi: 10.3934/dcdsb.2013.18.2143

[8]

Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601

[9]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993

[10]

Hirokazu Ninomiya, Masaharu Taniguchi. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 819-832. doi: 10.3934/dcds.2006.15.819

[11]

Yigui Ou, Yuanwen Liu. A memory gradient method based on the nonmonotone technique. Journal of Industrial & Management Optimization, 2017, 13 (2) : 857-872. doi: 10.3934/jimo.2016050

[12]

Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017

[13]

Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3145-3165. doi: 10.3934/dcdsb.2017168

[14]

Ahmet Sahiner, Nurullah Yilmaz, Gulden Kapusuz. A novel modeling and smoothing technique in global optimization. Journal of Industrial & Management Optimization, 2019, 15 (1) : 113-130. doi: 10.3934/jimo.2018035

[15]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325

[16]

Shiwang Ma, Xiao-Qiang Zhao. Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 259-275. doi: 10.3934/dcds.2008.21.259

[17]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727

[18]

Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 203-209. doi: 10.3934/dcdss.2020011

[19]

Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331

[20]

Cui-Ping Cheng, Wan-Tong Li, Zhi-Cheng Wang. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 559-575. doi: 10.3934/dcdsb.2010.13.559

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (18)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]