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Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations
Pushed traveling fronts in monostable equations with monotone delayed reaction
1. | Department of Differential Equations, National Technical University, Kyiv, Ukraine |
2. | Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile |
3. | Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca |
References:
[1] |
M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited), Math. Ann., 354 (2012), 73-109. |
[2] |
R. D. Benguria and M. C. Depassier, Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation, Comm. Math. Phys., 175 (1996), 221-227. |
[3] |
H. Berestycki and L. Nirenberg, Traveling waves in cylinders, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497-572. |
[4] |
H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844 . |
[5] |
A. Boumenir and V.M. Nguyen, Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Differential Equations, 244 (2008), 1551-1570. |
[6] |
A. Calamai, C. Marcelli and F. Papalini, A general approach for front-propagation in functional reaction-diffusion equations, J. Dynam. Differential Equations, 21 (2009) 567-392. |
[7] |
J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. |
[8] |
X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146. |
[9] |
X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258. |
[10] |
J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485. |
[11] |
J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118. |
[12] |
O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737. |
[13] |
U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Phys. D, 146 (2000), 1-99. |
[14] |
J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations, Proc. Amer. Math. Soc., 139 (2011), 1361-1373. |
[15] |
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. |
[16] |
T. Faria and S. Trofimchuk, Non-monotone traveling waves in a single species reaction,-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376. |
[17] |
B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction," Birkhauser, 2004. |
[18] |
C. Gomez, H. Prado and S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation,, preprint , ().
|
[19] |
K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263. |
[20] |
A. Kolmogorov, I. Petrovskii and N. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh., 1 (1937) 1-26. |
[21] |
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. Erratum: 61(2008), 137-138. |
[22] |
X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857-903. |
[23] |
S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. |
[24] |
S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277. |
[25] |
S. Ma, X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87. |
[26] |
J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-48. |
[27] |
M. Mei, Ch. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 233-258. |
[28] |
G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 553-557. |
[29] |
F. Rothe, Convergence to pushed fronts, Rocky Mountain J. Math. , 11 (1981), 617-633. |
[30] |
K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. |
[31] |
A.N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosciences, 31 (1976), 307-315. |
[32] |
K. Schumacher, Travelling-front solutions for integro-differential equations. I , J. Reine Angew. Math., 316 (1980), 54-70. |
[33] |
E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling wavefronts for a model of the Belousov-Zhabotinskii reaction,, preprint , ().
|
[34] |
E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction,, preprint , ().
|
[35] |
E. Trofimchuk, V. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay, J. Differential Equations, 245 (2008), 2307-2332. |
[36] |
E. Trofimchuk, P. Alvarado and S. Trofimchuk, On the geometry of wave solutions of a delayed reaction-diffusion equation, J. Differential Equations, 246 (2009), 1422-1444. |
[37] |
E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion equation with delay, Discrete Contin. Dyn. Syst., 20 (2008), 407-423. |
[38] |
Z.-C. Wang, W.T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607. |
[39] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. |
[40] |
P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. |
[41] |
J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687. |
[42] |
J. Xin}, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. |
[43] |
Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays, Proc. Amer. Math. Soc., 140 (2012), 3853-3859. |
[44] |
B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation J. Differential Equations, 105 (1993), 46-62. |
show all references
References:
[1] |
M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited), Math. Ann., 354 (2012), 73-109. |
[2] |
R. D. Benguria and M. C. Depassier, Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation, Comm. Math. Phys., 175 (1996), 221-227. |
[3] |
H. Berestycki and L. Nirenberg, Traveling waves in cylinders, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497-572. |
[4] |
H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844 . |
[5] |
A. Boumenir and V.M. Nguyen, Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Differential Equations, 244 (2008), 1551-1570. |
[6] |
A. Calamai, C. Marcelli and F. Papalini, A general approach for front-propagation in functional reaction-diffusion equations, J. Dynam. Differential Equations, 21 (2009) 567-392. |
[7] |
J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. |
[8] |
X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146. |
[9] |
X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258. |
[10] |
J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485. |
[11] |
J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118. |
[12] |
O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737. |
[13] |
U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Phys. D, 146 (2000), 1-99. |
[14] |
J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations, Proc. Amer. Math. Soc., 139 (2011), 1361-1373. |
[15] |
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. |
[16] |
T. Faria and S. Trofimchuk, Non-monotone traveling waves in a single species reaction,-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376. |
[17] |
B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction," Birkhauser, 2004. |
[18] |
C. Gomez, H. Prado and S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation,, preprint , ().
|
[19] |
K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263. |
[20] |
A. Kolmogorov, I. Petrovskii and N. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh., 1 (1937) 1-26. |
[21] |
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. Erratum: 61(2008), 137-138. |
[22] |
X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857-903. |
[23] |
S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. |
[24] |
S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277. |
[25] |
S. Ma, X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87. |
[26] |
J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-48. |
[27] |
M. Mei, Ch. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 233-258. |
[28] |
G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 553-557. |
[29] |
F. Rothe, Convergence to pushed fronts, Rocky Mountain J. Math. , 11 (1981), 617-633. |
[30] |
K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. |
[31] |
A.N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosciences, 31 (1976), 307-315. |
[32] |
K. Schumacher, Travelling-front solutions for integro-differential equations. I , J. Reine Angew. Math., 316 (1980), 54-70. |
[33] |
E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling wavefronts for a model of the Belousov-Zhabotinskii reaction,, preprint , ().
|
[34] |
E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction,, preprint , ().
|
[35] |
E. Trofimchuk, V. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay, J. Differential Equations, 245 (2008), 2307-2332. |
[36] |
E. Trofimchuk, P. Alvarado and S. Trofimchuk, On the geometry of wave solutions of a delayed reaction-diffusion equation, J. Differential Equations, 246 (2009), 1422-1444. |
[37] |
E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion equation with delay, Discrete Contin. Dyn. Syst., 20 (2008), 407-423. |
[38] |
Z.-C. Wang, W.T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607. |
[39] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. |
[40] |
P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. |
[41] |
J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687. |
[42] |
J. Xin}, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. |
[43] |
Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays, Proc. Amer. Math. Soc., 140 (2012), 3853-3859. |
[44] |
B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation J. Differential Equations, 105 (1993), 46-62. |
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