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Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay
1. | Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071, China |
2. | School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000 |
3. | Department of Applied Mathematics, Xidian University, Xi'an 710071, China |
References:
[1] |
J. Al-Omari and S. A. Gourley, Monotone traveling fronts in an age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294-312.
doi: 10.1007/s002850200159. |
[2] |
J. Al-Omari and S. A. Gourley, A nonlocal reaction-diffusion model for a single species with stage structure and distributed maturation delay, Euro. J. Appl. Math., 16 (2005), 37-51.
doi: 10.1017/S0956792504005716. |
[3] |
N. F. Britton, "Reaction-Diffusion Equations and Their Applications to Biology," Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. |
[4] |
X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. |
[5] |
S. A. Gourley, Linear stability of travelling fronts in an age-structured reaction-diffusion population model, Q. J. Mech. Appl. Math., 58 (2005), 257-268.
doi: 10.1093/qjmamj/hbi012. |
[6] |
G. Li, M. Mei and Y. Wong, Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model, Math. Biosci. Engin., 5 (2008), 85-100.
doi: 10.3934/mbe.2008.5.85. |
[7] |
W.-T. Li and S.-L. Wu, Traveling waves in a diffusive predator-prey model with Holling type-III functional response, Chaos, Solitons and Fractals, 37 (2008), 476-486.
doi: 10.1016/j.chaos.2006.09.039. |
[8] |
W.-T. Li, S. Ruan and Z.-C. Wang, On the diffusive Nicholson's blowflies equation with nonlocal delays, J. Nonlinear Sci., 17 (2007), 505-525.
doi: 10.1007/s00332-007-9003-9. |
[9] |
D. Liang and J. Wu, Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. Nonlinear Sci., 13 (2003), 289-310.
doi: 10.1007/s00332-003-0524-6. |
[10] |
C.-K. Lin and M. Mei, On Travelling wavefronts of the Nicholson's blowflies equation with diffusion, Proc. Royal Soc. Edinburgh A, 140 (2010), 135-152.
doi: 10.1017/S0308210508000784. |
[11] |
R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[12] |
S. Ma and 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. |
[13] |
M. Mei, J. W.-H. So, M. Li and S. Shen, Asymptotic stability of traveling waves for Nicholson's blowflies equation with diffusion, Proc. Royal Soc. Edinburgh A, 134 (2004), 579-594.
doi: 10.1017/S0308210500003358. |
[14] |
M. Mei and J. W.-H So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Royal Soc. Edinburgh A, 138 (2008), 551-568. |
[15] |
S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Royal Soc. Edinburgh A, 134 (2004), 991-1011.
doi: 10.1017/S0308210500003590. |
[16] |
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[17] |
K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. |
[18] |
H. L. Smith and X. Q. Zhao, Global asymptotical stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.
doi: 10.1137/S0036141098346785. |
[19] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, "Travelling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc., Providence, RI, 1994. |
[20] |
Z.-C. Wang, W.-T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. |
[21] |
Z.-C. Wang, W.-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. |
[22] |
Z.-C. Wang, W.-T. Li and S. Ruan, Travelling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 563-607. |
[23] |
J. Wu, "Theory and Applications of Partial Functional-Differential Equations,'' Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. |
[24] |
S.-L. Wu, W.-T. Li and S.-Y. Liu, Oscillatory waves in reaction-diffusion equations with nonlocal delay and crossing-monostability, Nonlinear Anal. RWA, 10 (2009), 3141-3151.
doi: 10.1016/j.nonrwa.2008.10.012. |
[25] |
S.-L. Wu, W.-T. Li and S.-Y. Liu, Asymptotic stability of traveling wave fronts in nonlocal reaction-diffusion equations with delay, J. Math. Anal. Appl., 360 (2009), 439-458.
doi: 10.1016/j.jmaa.2009.06.061. |
[26] |
S.-L. Wu, H.-Q. Zhao and S.-Y. Liu, Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability, Z. angew. Math. Phys., 62 (2011), 377-397.
doi: 10.1007/s00033-010-0112-1. |
[27] |
J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
[28] |
Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Science Press, Beijing, 1990. |
show all references
References:
[1] |
J. Al-Omari and S. A. Gourley, Monotone traveling fronts in an age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294-312.
doi: 10.1007/s002850200159. |
[2] |
J. Al-Omari and S. A. Gourley, A nonlocal reaction-diffusion model for a single species with stage structure and distributed maturation delay, Euro. J. Appl. Math., 16 (2005), 37-51.
doi: 10.1017/S0956792504005716. |
[3] |
N. F. Britton, "Reaction-Diffusion Equations and Their Applications to Biology," Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. |
[4] |
X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. |
[5] |
S. A. Gourley, Linear stability of travelling fronts in an age-structured reaction-diffusion population model, Q. J. Mech. Appl. Math., 58 (2005), 257-268.
doi: 10.1093/qjmamj/hbi012. |
[6] |
G. Li, M. Mei and Y. Wong, Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model, Math. Biosci. Engin., 5 (2008), 85-100.
doi: 10.3934/mbe.2008.5.85. |
[7] |
W.-T. Li and S.-L. Wu, Traveling waves in a diffusive predator-prey model with Holling type-III functional response, Chaos, Solitons and Fractals, 37 (2008), 476-486.
doi: 10.1016/j.chaos.2006.09.039. |
[8] |
W.-T. Li, S. Ruan and Z.-C. Wang, On the diffusive Nicholson's blowflies equation with nonlocal delays, J. Nonlinear Sci., 17 (2007), 505-525.
doi: 10.1007/s00332-007-9003-9. |
[9] |
D. Liang and J. Wu, Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. Nonlinear Sci., 13 (2003), 289-310.
doi: 10.1007/s00332-003-0524-6. |
[10] |
C.-K. Lin and M. Mei, On Travelling wavefronts of the Nicholson's blowflies equation with diffusion, Proc. Royal Soc. Edinburgh A, 140 (2010), 135-152.
doi: 10.1017/S0308210508000784. |
[11] |
R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[12] |
S. Ma and 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. |
[13] |
M. Mei, J. W.-H. So, M. Li and S. Shen, Asymptotic stability of traveling waves for Nicholson's blowflies equation with diffusion, Proc. Royal Soc. Edinburgh A, 134 (2004), 579-594.
doi: 10.1017/S0308210500003358. |
[14] |
M. Mei and J. W.-H So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Royal Soc. Edinburgh A, 138 (2008), 551-568. |
[15] |
S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Royal Soc. Edinburgh A, 134 (2004), 991-1011.
doi: 10.1017/S0308210500003590. |
[16] |
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[17] |
K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. |
[18] |
H. L. Smith and X. Q. Zhao, Global asymptotical stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.
doi: 10.1137/S0036141098346785. |
[19] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, "Travelling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc., Providence, RI, 1994. |
[20] |
Z.-C. Wang, W.-T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. |
[21] |
Z.-C. Wang, W.-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. |
[22] |
Z.-C. Wang, W.-T. Li and S. Ruan, Travelling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 563-607. |
[23] |
J. Wu, "Theory and Applications of Partial Functional-Differential Equations,'' Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. |
[24] |
S.-L. Wu, W.-T. Li and S.-Y. Liu, Oscillatory waves in reaction-diffusion equations with nonlocal delay and crossing-monostability, Nonlinear Anal. RWA, 10 (2009), 3141-3151.
doi: 10.1016/j.nonrwa.2008.10.012. |
[25] |
S.-L. Wu, W.-T. Li and S.-Y. Liu, Asymptotic stability of traveling wave fronts in nonlocal reaction-diffusion equations with delay, J. Math. Anal. Appl., 360 (2009), 439-458.
doi: 10.1016/j.jmaa.2009.06.061. |
[26] |
S.-L. Wu, H.-Q. Zhao and S.-Y. Liu, Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability, Z. angew. Math. Phys., 62 (2011), 377-397.
doi: 10.1007/s00033-010-0112-1. |
[27] |
J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
[28] |
Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Science Press, Beijing, 1990. |
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