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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), 294-312. 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), 37-51. doi: 10.1017/S0956792504005716.  Google Scholar

[3]

N. F. Britton, "Reaction-Diffusion Equations and Their Applications to Biology," Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986.  Google Scholar

[4]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.  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), 257-268. 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), 85-100. 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, Solitons and Fractals, 37 (2008), 476-486. 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), 505-525. 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), 289-310. 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), 135-152. 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), 1-44. 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), 54-87.  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), 579-594. 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), 551-568. 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), 991-1011. doi: 10.1017/S0308210500003590.  Google Scholar

[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.  Google Scholar

[17]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  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), 514-534. 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, Amer. Math. Soc., Providence, RI, 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), 185-232. 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), 153-200.  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), 563-607. Google Scholar

[23]

J. Wu, "Theory and Applications of Partial Functional-Differential Equations,'' Applied Mathematical Sciences, 119, Springer-Verlag, New York, 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), 3141-3151. 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), 439-458. 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), 377-397. doi: 10.1007/s00033-010-0112-1.  Google Scholar

[27]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.  Google Scholar

[28]

Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Science Press, Beijing, 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), 294-312. 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), 37-51. doi: 10.1017/S0956792504005716.  Google Scholar

[3]

N. F. Britton, "Reaction-Diffusion Equations and Their Applications to Biology," Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986.  Google Scholar

[4]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.  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), 257-268. 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), 85-100. 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, Solitons and Fractals, 37 (2008), 476-486. 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), 505-525. 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), 289-310. 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), 135-152. 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), 1-44. 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), 54-87.  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), 579-594. 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), 551-568. 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), 991-1011. doi: 10.1017/S0308210500003590.  Google Scholar

[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.  Google Scholar

[17]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  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), 514-534. 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, Amer. Math. Soc., Providence, RI, 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), 185-232. 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), 153-200.  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), 563-607. Google Scholar

[23]

J. Wu, "Theory and Applications of Partial Functional-Differential Equations,'' Applied Mathematical Sciences, 119, Springer-Verlag, New York, 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), 3141-3151. 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), 439-458. 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), 377-397. doi: 10.1007/s00033-010-0112-1.  Google Scholar

[27]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.  Google Scholar

[28]

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

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