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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

 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.
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. [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. [3] N. F. Britton, "Reaction-Diffusion Equations and Their Applications to Biology,", Academic Press, (1986). [4] X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587. [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. [19] A. I. Volpert, V. A. Volpert and V. A. Volpert, "Travelling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs,, Vol. 140, (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. [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. [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. [23] J. Wu, "Theory and Applications of Partial Functional-Differential Equations,'', Applied Mathematical Sciences, 119 (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. 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. 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. doi: 10.1007/s00033-010-0112-1. [27] J. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161. doi: 10.1137/S0036144599364296. [28] Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations,", Science Press, (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. 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. doi: 10.1017/S0956792504005716. [3] N. F. Britton, "Reaction-Diffusion Equations and Their Applications to Biology,", Academic Press, (1986). [4] X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587. [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. [19] A. I. Volpert, V. A. Volpert and V. A. Volpert, "Travelling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs,, Vol. 140, (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. [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. [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. [23] J. Wu, "Theory and Applications of Partial Functional-Differential Equations,'', Applied Mathematical Sciences, 119 (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. 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. 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. doi: 10.1007/s00033-010-0112-1. [27] J. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161. doi: 10.1137/S0036144599364296. [28] Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations,", Science Press, (1990).
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