American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stability and Hopf bifurcations for a delayed diffusion system in population dynamics
  • DCDS-B Home
  • This Issue
  • Next Article
    Generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions in unbounded domains
January  2012, 17(1): 347-366. doi: 10.3934/dcdsb.2012.17.347

Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay

Shi-Liang Wu 1, , Wan-Tong Li 2, and  San-Yang Liu 3,

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

Received  June 2009 Revised  May 2011 Published  October 2011

This paper is concerned with the exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with nonlocal delay. The existence and comparison theorem of solutions of the corresponding Cauchy problem in a weighted Sobolev space are first established for the systems on $\mathbb{R}$ by appealing to the theory of semigroup and abstract functional differential equations. The exponential stability of traveling fronts is then proved by the comparison principle and the (technical) weighted energy method. Comparing with the previous results, our results recovers and/or improves a number of existing ones. Finally, we apply our results to some biological and epidemic models and obtain some new results.
Keywords: comparison method, weighted energy method, non-local delay, Exponential stability, monostable., traveling front, reaction-advection-diffusion equation.
Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 35R20, 92D2.
Citation: Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347
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.  doi: 10.1007/s002850200159.  Google Scholar

[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.  doi: 10.1017/S0956792504005716.  Google Scholar

[3]

N. F. Britton, "Reaction-Diffusion Equations and Their Applications to Biology,", Academic Press, (1986).   Google Scholar

[4]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125.   Google Scholar

[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.  doi: 10.1093/qjmamj/hbi012.  Google Scholar

[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.  doi: 10.3934/mbe.2008.5.85.  Google Scholar

[7]

W.-T. Li and S.-L. Wu, Traveling waves in a diffusive predator-prey model with Holling type-III functional response,, Chaos, 37 (2008), 476.  doi: 10.1016/j.chaos.2006.09.039.  Google Scholar

[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.  doi: 10.1007/s00332-007-9003-9.  Google Scholar

[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.  doi: 10.1007/s00332-003-0524-6.  Google Scholar

[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.  doi: 10.1017/S0308210508000784.  Google Scholar

[11]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1.  doi: 10.2307/2001590.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1017/S0308210500003358.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1017/S0308210500003590.  Google Scholar

[16]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems,, Adv. Math., 22 (1976), 312.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[17]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587.   Google Scholar

[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.  doi: 10.1137/S0036141098346785.  Google Scholar

[19]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Travelling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs,, Vol. 140, (1994).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[23]

J. Wu, "Theory and Applications of Partial Functional-Differential Equations,'', Applied Mathematical Sciences, 119 (1996).   Google Scholar

[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.  doi: 10.1016/j.nonrwa.2008.10.012.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2009.06.061.  Google Scholar

[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.  doi: 10.1007/s00033-010-0112-1.  Google Scholar

[27]

J. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161.  doi: 10.1137/S0036144599364296.  Google Scholar

[28]

Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations,", Science Press, (1990).   Google Scholar

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.  doi: 10.1007/s002850200159.  Google Scholar

[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.  doi: 10.1017/S0956792504005716.  Google Scholar

[3]

N. F. Britton, "Reaction-Diffusion Equations and Their Applications to Biology,", Academic Press, (1986).   Google Scholar

[4]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125.   Google Scholar

[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.  doi: 10.1093/qjmamj/hbi012.  Google Scholar

[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.  doi: 10.3934/mbe.2008.5.85.  Google Scholar

[7]

W.-T. Li and S.-L. Wu, Traveling waves in a diffusive predator-prey model with Holling type-III functional response,, Chaos, 37 (2008), 476.  doi: 10.1016/j.chaos.2006.09.039.  Google Scholar

[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.  doi: 10.1007/s00332-007-9003-9.  Google Scholar

[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.  doi: 10.1007/s00332-003-0524-6.  Google Scholar

[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.  doi: 10.1017/S0308210508000784.  Google Scholar

[11]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1.  doi: 10.2307/2001590.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1017/S0308210500003358.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1017/S0308210500003590.  Google Scholar

[16]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems,, Adv. Math., 22 (1976), 312.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[17]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587.   Google Scholar

[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.  doi: 10.1137/S0036141098346785.  Google Scholar

[19]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Travelling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs,, Vol. 140, (1994).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[23]

J. Wu, "Theory and Applications of Partial Functional-Differential Equations,'', Applied Mathematical Sciences, 119 (1996).   Google Scholar

[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.  doi: 10.1016/j.nonrwa.2008.10.012.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2009.06.061.  Google Scholar

[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.  doi: 10.1007/s00033-010-0112-1.  Google Scholar

[27]

J. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161.  doi: 10.1137/S0036144599364296.  Google Scholar

[28]

Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations,", Science Press, (1990).   Google Scholar

[1]

Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255
[2]

Zhen-Hui Bu, Zhi-Cheng Wang. Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media. Communications on Pure & Applied Analysis, 2016, 15 (1) : 139-160. doi: 10.3934/cpaa.2016.15.139
[3]

Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
[4]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029
[5]

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
[6]

Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115
[7]

Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029
[8]

Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71
[9]

Mihaela Negreanu, J. Ignacio Tello. On a comparison method to reaction-diffusion systems and its applications to chemotaxis. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2669-2688. doi: 10.3934/dcdsb.2013.18.2669
[10]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147
[11]

Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3145-3165. doi: 10.3934/dcdsb.2017168
[12]

Antonio Greco, Vincenzino Mascia. Non-local sublinear problems: Existence, comparison, and radial symmetry. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 503-519. doi: 10.3934/dcds.2019021
[13]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[14]

Alexandre Caboussat, Roland Glowinski. A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics. Communications on Pure & Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161
[15]

Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037
[16]

Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203
[17]

Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks & Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002
[18]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465
[19]

Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681
[20]

Alexey Penenko. Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements. Inverse Problems & Imaging, 2020, 14 (5) : 757-782. doi: 10.3934/ipi.2020035

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences