American Institute of Mathematical Sciences

Advanced
March  2016, 15(2): 319-334. doi: 10.3934/cpaa.2016.15.319

Traveling wave solutions of a reaction-diffusion equation with state-dependent delay

Guo Lin 1, and  Haiyan Wang 2,

1. 

School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000
2. 

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

Received  October 2014 Revised  October 2015 Published  January 2016

This paper is concerned with the traveling wave solutions of a reaction-diffusion equation with state-dependent delay. When the birth function is monotone, the existence and nonexistence of monotone traveling wave solutions are established. When the birth function is not monotone, the minimal wave speed of nontrivial traveling wave solutions is obtained. The results are proved by the construction of upper and lower solutions and application of the fixed point theorem.
Keywords: asymptotic spreading, minimal wave speed., Comparison principle.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35C07; 37C6.
Citation: Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319
References:
[1]

W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay,, \emph{SIAM J. Appl. Math.}, 52 (1982), 855.  doi: 10.1137/0152048.  Google Scholar

[2]

H. G. Andrewartha and L. C. Birch, The Distribution and Abundance of Animals,, University of Chicago Press, (1954).   Google Scholar

[3]

O. Arino, K. P. Hadeler and M. L. Hbid, Existence of periodic solutions for delay differential equations with state dependent delay,, \emph{J. Differential Equations}, 144 (1998), 263.  doi: 10.1006/jdeq.1997.3378.  Google Scholar

[4]

K. L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations,, \emph{Proc. Amer. Math. Soc.}, 124 (1996), 1417.  doi: 10.1090/S0002-9939-96-03437-5.  Google Scholar

[5]

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

[6]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications,, in \emph{Handbook of Differential Equations: Ordinary Differential Equations} (eds. A. Canada), (2006), 435.   Google Scholar

[7]

Q. Hu, J. Wu and X. Zou, Estimates of periods and global continua of periodic solutions for state-dependent delay equations,, \emph{SIAM J. Math. Anal., 44 (2012), 2401.  doi: 10.1137/100793712.  Google Scholar

[8]

X. Liang and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, \emph{Comm. Pure Appl. Math., 60 (2007), 1.  doi: 10.1002/cpa.20154.  Google Scholar

[9]

G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays,, \emph{J. Dynam. Diff. Eqns., 26 (2014), 583.  doi: 10.1007/s10884-014-9355-4.  Google Scholar

[10]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, \emph{J. Differential Equations, 171 (2001), 294.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[11]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, \emph{J. Differential Equations, 237 (2007), 259.  doi: 10.1016/j.jde.2007.03.014.  Google Scholar

[12]

P. Magal and O. Arino, Existence of periodic solutions for a state-dependent delay differential equation,, \emph{J. Differential Equations, 165 (2000), 61.  doi: 10.1006/jdeq.1999.3759.  Google Scholar

[13]

J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations,, \emph{J. Differential Equations, 250 (2011), 4037.  doi: 10.1016/j.jde.2010.10.024.  Google Scholar

[14]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations,, \emph{Trans. Amer. Math. Soc., 302 (1987), 587.  doi: 10.2307/2000859 .  Google Scholar

[15]

H. L. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations,, \emph{SIAM J. Math. Anal., 31 (2000), 514.  doi: 10.1137/S0036141098346785.  Google Scholar

[16]

H. R. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models,, \emph{J. Differential Equations, 195 (2003), 430.  doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar

[17]

H. O. Walther, Semiflows for neutral equations with state-dependent delays,, in \emph{Infinite dimensional dynamical systems} (eds. J. Mallet-Paret, (2013), 211.   Google Scholar

[18]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, \emph{J. Differential Equations, 247 (2009), 887.  doi: 10.1016/j.jde.2009.04.002.  Google Scholar

[19]

Z. C. Wang, W. T. Li and S. Ruan, Traveling wave fronts of reaction-diffusion systems with spatio-temporal delays,, \emph{J. Differential Equations, 222 (2006), 185.  doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[20]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, \emph{J. Dynam. Diff. Eqns., 13 (2001), 651.  doi: 10.1023/A:1016690424892.  Google Scholar

show all references

References:
[1]

W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay,, \emph{SIAM J. Appl. Math.}, 52 (1982), 855.  doi: 10.1137/0152048.  Google Scholar

[2]

H. G. Andrewartha and L. C. Birch, The Distribution and Abundance of Animals,, University of Chicago Press, (1954).   Google Scholar

[3]

O. Arino, K. P. Hadeler and M. L. Hbid, Existence of periodic solutions for delay differential equations with state dependent delay,, \emph{J. Differential Equations}, 144 (1998), 263.  doi: 10.1006/jdeq.1997.3378.  Google Scholar

[4]

K. L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations,, \emph{Proc. Amer. Math. Soc.}, 124 (1996), 1417.  doi: 10.1090/S0002-9939-96-03437-5.  Google Scholar

[5]

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

[6]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications,, in \emph{Handbook of Differential Equations: Ordinary Differential Equations} (eds. A. Canada), (2006), 435.   Google Scholar

[7]

Q. Hu, J. Wu and X. Zou, Estimates of periods and global continua of periodic solutions for state-dependent delay equations,, \emph{SIAM J. Math. Anal., 44 (2012), 2401.  doi: 10.1137/100793712.  Google Scholar

[8]

X. Liang and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, \emph{Comm. Pure Appl. Math., 60 (2007), 1.  doi: 10.1002/cpa.20154.  Google Scholar

[9]

G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays,, \emph{J. Dynam. Diff. Eqns., 26 (2014), 583.  doi: 10.1007/s10884-014-9355-4.  Google Scholar

[10]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, \emph{J. Differential Equations, 171 (2001), 294.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[11]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, \emph{J. Differential Equations, 237 (2007), 259.  doi: 10.1016/j.jde.2007.03.014.  Google Scholar

[12]

P. Magal and O. Arino, Existence of periodic solutions for a state-dependent delay differential equation,, \emph{J. Differential Equations, 165 (2000), 61.  doi: 10.1006/jdeq.1999.3759.  Google Scholar

[13]

J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations,, \emph{J. Differential Equations, 250 (2011), 4037.  doi: 10.1016/j.jde.2010.10.024.  Google Scholar

[14]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations,, \emph{Trans. Amer. Math. Soc., 302 (1987), 587.  doi: 10.2307/2000859 .  Google Scholar

[15]

H. L. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations,, \emph{SIAM J. Math. Anal., 31 (2000), 514.  doi: 10.1137/S0036141098346785.  Google Scholar

[16]

H. R. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models,, \emph{J. Differential Equations, 195 (2003), 430.  doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar

[17]

H. O. Walther, Semiflows for neutral equations with state-dependent delays,, in \emph{Infinite dimensional dynamical systems} (eds. J. Mallet-Paret, (2013), 211.   Google Scholar

[18]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, \emph{J. Differential Equations, 247 (2009), 887.  doi: 10.1016/j.jde.2009.04.002.  Google Scholar

[19]

Z. C. Wang, W. T. Li and S. Ruan, Traveling wave fronts of reaction-diffusion systems with spatio-temporal delays,, \emph{J. Differential Equations, 222 (2006), 185.  doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[20]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, \emph{J. Dynam. Diff. Eqns., 13 (2001), 651.  doi: 10.1023/A:1016690424892.  Google Scholar

[1]

Shuxia Pan. Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle. Electronic Research Archive, 2019, 27: 89-99. doi: 10.3934/era.2019011
[2]

Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213
[3]

Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067
[4]

Hans Weinberger. On sufficient conditions for a linearly determinate spreading speed. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2267-2280. doi: 10.3934/dcdsb.2012.17.2267
[5]

Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks & Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583
[6]

Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69
[7]

Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591
[8]

Meng Zhao, Wan-Tong Li, Wenjie Ni. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 981-999. doi: 10.3934/dcdsb.2019199
[9]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769
[10]

Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116
[11]

Shigeaki Koike, Takahiro Kosugi. Remarks on the comparison principle for quasilinear PDE with no zeroth order terms. Communications on Pure & Applied Analysis, 2015, 14 (1) : 133-142. doi: 10.3934/cpaa.2015.14.133
[12]

Xiaowei Tang, Xilin Fu. New comparison principle with Razumikhin condition for impulsive infinite delay differential systems. Conference Publications, 2009, 2009 (Special) : 739-743. doi: 10.3934/proc.2009.2009.739
[13]

Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic & Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725
[14]

Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897
[15]

Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026
[16]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047
[17]

Peixuan Weng. Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 883-904. doi: 10.3934/dcdsb.2009.12.883
[18]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785
[19]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441
[20]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences