    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

 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.
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:
  W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1982), 855-869. doi: 10.1137/0152048.  Google Scholar  H. G. Andrewartha and L. C. Birch, The Distribution and Abundance of Animals, University of Chicago Press, Chicago, IL, 1954. Google Scholar  O. Arino, K. P. Hadeler and M. L. Hbid, Existence of periodic solutions for delay differential equations with state dependent delay, J. Differential Equations, 144 (1998), 263-301. doi: 10.1006/jdeq.1997.3378.  Google Scholar  K. L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 1417-1426. doi: 10.1090/S0002-9939-96-03437-5.  Google Scholar  J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009.  Google Scholar  F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in Handbook of Differential Equations: Ordinary Differential Equations (eds. A. Canada), Elsevier, Dordrecht, The Netherlands, (2006), 435-545. Google Scholar  Q. Hu, J. Wu and X. Zou, Estimates of periods and global continua of periodic solutions for state-dependent delay equations, SIAM J. Math. Anal., 44 (2012), 2401-2427. doi: 10.1137/100793712.  Google Scholar  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.  Google Scholar  G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays, J. Dynam. Diff. Eqns., 26 (2014), 583-605. doi: 10.1007/s10884-014-9355-4.  Google Scholar  S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846.  Google Scholar  S. 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.  Google Scholar  P. Magal and O. Arino, Existence of periodic solutions for a state-dependent delay differential equation, J. Differential Equations, 165 (2000), 61-95. doi: 10.1006/jdeq.1999.3759.  Google Scholar  J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations, J. Differential Equations, 250 (2011), 4037-4084. doi: 10.1016/j.jde.2010.10.024.  Google Scholar  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 .  Google Scholar  H. L. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785.  Google Scholar  H. R. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar  H. O. Walther, Semiflows for neutral equations with state-dependent delays, in Infinite dimensional dynamical systems (eds. J. Mallet-Paret, J. Wu, Y. Yi and H. Zhu), Fields Inst. Commun., 64, Springer, New York, (2013), 211-267. Google Scholar  H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations, J. Differential Equations, 247 (2009), 887-905. doi: 10.1016/j.jde.2009.04.002.  Google Scholar  Z. C. Wang, W. T. Li and S. Ruan, Traveling wave fronts of reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010.  Google Scholar  J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Diff. Eqns., 13 (2001), 651-687. doi: 10.1023/A:1016690424892.  Google Scholar

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##### References:
  W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1982), 855-869. doi: 10.1137/0152048.  Google Scholar  H. G. Andrewartha and L. C. Birch, The Distribution and Abundance of Animals, University of Chicago Press, Chicago, IL, 1954. Google Scholar  O. Arino, K. P. Hadeler and M. L. Hbid, Existence of periodic solutions for delay differential equations with state dependent delay, J. Differential Equations, 144 (1998), 263-301. doi: 10.1006/jdeq.1997.3378.  Google Scholar  K. L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 1417-1426. doi: 10.1090/S0002-9939-96-03437-5.  Google Scholar  J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009.  Google Scholar  F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in Handbook of Differential Equations: Ordinary Differential Equations (eds. A. Canada), Elsevier, Dordrecht, The Netherlands, (2006), 435-545. Google Scholar  Q. Hu, J. Wu and X. Zou, Estimates of periods and global continua of periodic solutions for state-dependent delay equations, SIAM J. Math. Anal., 44 (2012), 2401-2427. doi: 10.1137/100793712.  Google Scholar  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.  Google Scholar  G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays, J. Dynam. Diff. Eqns., 26 (2014), 583-605. doi: 10.1007/s10884-014-9355-4.  Google Scholar  S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846.  Google Scholar  S. 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.  Google Scholar  P. Magal and O. Arino, Existence of periodic solutions for a state-dependent delay differential equation, J. Differential Equations, 165 (2000), 61-95. doi: 10.1006/jdeq.1999.3759.  Google Scholar  J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations, J. Differential Equations, 250 (2011), 4037-4084. doi: 10.1016/j.jde.2010.10.024.  Google Scholar  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 .  Google Scholar  H. L. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785.  Google Scholar  H. R. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar  H. O. Walther, Semiflows for neutral equations with state-dependent delays, in Infinite dimensional dynamical systems (eds. J. Mallet-Paret, J. Wu, Y. Yi and H. Zhu), Fields Inst. Commun., 64, Springer, New York, (2013), 211-267. Google Scholar  H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations, J. Differential Equations, 247 (2009), 887-905. doi: 10.1016/j.jde.2009.04.002.  Google Scholar  Z. C. Wang, W. T. Li and S. Ruan, Traveling wave fronts of reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010.  Google Scholar  J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Diff. Eqns., 13 (2001), 651-687. doi: 10.1023/A:1016690424892.  Google Scholar

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