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On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems
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 |
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, SIAM J. Appl. Math., 52 (1982), 855-869.
doi: 10.1137/0152048. |
[2] |
H. G. Andrewartha and L. C. Birch, The Distribution and Abundance of Animals, University of Chicago Press, Chicago, IL, 1954. |
[3] |
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. |
[4] |
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. |
[5] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
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. |
[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, J. Dynam. Diff. Eqns., 26 (2014), 583-605.
doi: 10.1007/s10884-014-9355-4. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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 . |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
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, SIAM J. Appl. Math., 52 (1982), 855-869.
doi: 10.1137/0152048. |
[2] |
H. G. Andrewartha and L. C. Birch, The Distribution and Abundance of Animals, University of Chicago Press, Chicago, IL, 1954. |
[3] |
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. |
[4] |
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. |
[5] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
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. |
[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, J. Dynam. Diff. Eqns., 26 (2014), 583-605.
doi: 10.1007/s10884-014-9355-4. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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 . |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
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