# American Institute of Mathematical Sciences

March  2016, 36(3): 1331-1353. doi: 10.3934/dcds.2016.36.1331

## Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations

 1 School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong, 510631 2 Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2 3 School of Mathematics and Statistics, Northeast Normal University, Changchun, MO 130024 4 School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, China

Received  December 2014 Revised  March 2015 Published  August 2015

This paper is concerned with the stability of non-monotone traveling waves to a nonlocal dispersion equation with time-delay, a time-delayed integro-differential equation. When the equation is crossing-monostable, the equation and the traveling waves both loss their monotonicity, and the traveling waves are oscillating as the time-delay is big. In this paper, we prove that all non-critical traveling waves (the wave speed is greater than the minimum speed), including those oscillatory waves, are time-exponentially stable, when the initial perturbations around the waves are small. The adopted approach is still the technical weighted-energy method but with a new development. Numerical simulations in different cases are also carried out, which further confirm our theoretical result. Finally, as a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves for the non-monotone integro-differential equation, which was open so far as we know.
Citation: Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331
##### References:
 [1] M. Aguerra, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts, Math. Ann., 354 (2012), 73-109. doi: 10.1007/s00208-011-0722-8. [2] F. Andreu-Vaillo, J. M. Mazon, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Math. Surveys and Monographs, Vol. 165, Amer. Math. Soc., 2010. doi: 10.1090/surv/165. [3] E. Chasseigne, M. Chaves and J. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pure Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [4] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Rational Mech. Anal., 187 (2008), 137-156. doi: 10.1007/s00205-007-0062-8. [5] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction-diffusion equation, Annali. di Matematica Pura Appl., 185 (2006), 461-485. doi: 10.1007/s10231-005-0163-7. [6] J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002. [7] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721. [8] J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030. [9] 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. doi: 10.1016/j.jde.2010.01.009. [10] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979. [11] W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0. [12] A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. London Math. Soc., 89 (2014), 47-68. doi: 10.1112/jlms/jdt050. [13] S. A. Gourley, J. W.-H. So and J. Wu, Nonlocalily of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics, (Russian) Sovrem. Mat. Fundam. Napravl., 1 (2003), 84-120; translation in J. Math. Sci., 124 (2004), 5119-5153. doi: 10.1023/B:JOTH.0000047249.39572.6d. [14] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations (eds. H. Brunner, X.-Q. Zhao and X. Zou), Fields Institute Communications, 48, Amer. Math. Soc., Providence, RI, 2006, 137-200. [15] R. Huang, M. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discret. Contin. Dyn. Syst. A, 32 (2012), 3621-3649. doi: 10.3934/dcds.2012.32.3621. [16] L. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pure Appl.(9), 92 (2009), 163-187. doi: 10.1016/j.matpur.2009.04.009. [17] L. Ignat and J. D. Rossi, A nonlocal convolution-diffusion equation, J. Func. Anal., 251 (2007), 399-437. doi: 10.1016/j.jfa.2007.07.013. [18] 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. [19] C.-K. Lin, C.-T. Lin, Y. P. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084. doi: 10.1137/120904391. [20] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. [21] A. Matsumura and M. Mei, Nonlinear stability of viscous shock profile for a non-convex system of viscoelasticity, Osaka J. Math., 34 (1997), 589-603. [22] M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026. [23] M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020. [24] M. Mei, J. W.-H. So, M. Li and S. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sec. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358. [25] M. Mei and J. W.-H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sec. A, 138 (2008), 551-568. doi: 10.1017/S0308210506000333. [26] M. Mei, C. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790; Erratum, SIAM J. Math. Anal., 44 (2012), 538-540. doi: 10.1137/090776342. [27] M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Numer. Anal. Model. Seris B, 2 (2011), 379-401. [28] M. Mei and Y. S. Wong, Novel stability results for traveling wavefronts in an age-structured reaction-diffusion equations, Math. Biosci. Engin., 6 (2009), 743-752. doi: 10.3934/mbe.2009.6.743. [29] J. A. J. Metz and O. Diekmann, The dynamics of Physiologically Structured Populations, Springer, New York, 1986. doi: 10.1007/978-3-662-13159-6. [30] H. J. K. Moet, A note on the asymptotic behavior of solutions of the KPP equation, SIAM J. Math. Anal., 10 (1979), 728-732. doi: 10.1137/0510067. [31] S. Pan, W.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158. doi: 10.1016/j.na.2009.12.008. [32] J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure: I. Traveling wavefronts on unbounded domains, Roy. Soc. London Proc. Series A Math. Phys. Eng. Sci., 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789. [33] E. Trofimchuk and S. Trofimchunk, Admissible wavefront speeds for a single species reaction-diffusion equation with delay, Discrete Contin. Dyn. Syst. A, 20 (2008), 407-423. doi: 10.3934/dcds.2008.20.407. [34] 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. doi: 10.1016/j.jde.2008.06.023. [35] 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. [36] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953. doi: 10.2977/prims/1260476648. [37] G.-B. Zhang, Traveling waves in a nonlocal dispersal population model with age-structure, Nonlinear Anal., 74 (2011), 5030-5047. doi: 10.1016/j.na.2011.04.069. [38] G.-B. Zhang and R. Ma, Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, Z. Ang. Math. Phys., 64 (2013), 1643-1659. doi: 10.1007/s00033-013-0353-x.

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##### References:
 [1] M. Aguerra, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts, Math. Ann., 354 (2012), 73-109. doi: 10.1007/s00208-011-0722-8. [2] F. Andreu-Vaillo, J. M. Mazon, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Math. Surveys and Monographs, Vol. 165, Amer. Math. Soc., 2010. doi: 10.1090/surv/165. [3] E. Chasseigne, M. Chaves and J. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pure Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [4] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Rational Mech. Anal., 187 (2008), 137-156. doi: 10.1007/s00205-007-0062-8. [5] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction-diffusion equation, Annali. di Matematica Pura Appl., 185 (2006), 461-485. doi: 10.1007/s10231-005-0163-7. [6] J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002. [7] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721. [8] J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030. [9] 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. doi: 10.1016/j.jde.2010.01.009. [10] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979. [11] W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0. [12] A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. London Math. Soc., 89 (2014), 47-68. doi: 10.1112/jlms/jdt050. [13] S. A. Gourley, J. W.-H. So and J. Wu, Nonlocalily of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics, (Russian) Sovrem. Mat. Fundam. Napravl., 1 (2003), 84-120; translation in J. Math. Sci., 124 (2004), 5119-5153. doi: 10.1023/B:JOTH.0000047249.39572.6d. [14] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations (eds. H. Brunner, X.-Q. Zhao and X. Zou), Fields Institute Communications, 48, Amer. Math. Soc., Providence, RI, 2006, 137-200. [15] R. Huang, M. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discret. Contin. Dyn. Syst. A, 32 (2012), 3621-3649. doi: 10.3934/dcds.2012.32.3621. [16] L. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pure Appl.(9), 92 (2009), 163-187. doi: 10.1016/j.matpur.2009.04.009. [17] L. Ignat and J. D. Rossi, A nonlocal convolution-diffusion equation, J. Func. Anal., 251 (2007), 399-437. doi: 10.1016/j.jfa.2007.07.013. [18] 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. [19] C.-K. Lin, C.-T. Lin, Y. P. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084. doi: 10.1137/120904391. [20] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. [21] A. Matsumura and M. Mei, Nonlinear stability of viscous shock profile for a non-convex system of viscoelasticity, Osaka J. Math., 34 (1997), 589-603. [22] M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026. [23] M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020. [24] M. Mei, J. W.-H. So, M. Li and S. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sec. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358. [25] M. Mei and J. W.-H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sec. A, 138 (2008), 551-568. doi: 10.1017/S0308210506000333. [26] M. Mei, C. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790; Erratum, SIAM J. Math. Anal., 44 (2012), 538-540. doi: 10.1137/090776342. [27] M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Numer. Anal. Model. Seris B, 2 (2011), 379-401. [28] M. Mei and Y. S. Wong, Novel stability results for traveling wavefronts in an age-structured reaction-diffusion equations, Math. Biosci. Engin., 6 (2009), 743-752. doi: 10.3934/mbe.2009.6.743. [29] J. A. J. Metz and O. Diekmann, The dynamics of Physiologically Structured Populations, Springer, New York, 1986. doi: 10.1007/978-3-662-13159-6. [30] H. J. K. Moet, A note on the asymptotic behavior of solutions of the KPP equation, SIAM J. Math. Anal., 10 (1979), 728-732. doi: 10.1137/0510067. [31] S. Pan, W.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158. doi: 10.1016/j.na.2009.12.008. [32] J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure: I. Traveling wavefronts on unbounded domains, Roy. Soc. London Proc. Series A Math. Phys. Eng. Sci., 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789. [33] E. Trofimchuk and S. Trofimchunk, Admissible wavefront speeds for a single species reaction-diffusion equation with delay, Discrete Contin. Dyn. Syst. A, 20 (2008), 407-423. doi: 10.3934/dcds.2008.20.407. [34] 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. doi: 10.1016/j.jde.2008.06.023. [35] 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. [36] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953. doi: 10.2977/prims/1260476648. [37] G.-B. Zhang, Traveling waves in a nonlocal dispersal population model with age-structure, Nonlinear Anal., 74 (2011), 5030-5047. doi: 10.1016/j.na.2011.04.069. [38] G.-B. Zhang and R. Ma, Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, Z. Ang. Math. Phys., 64 (2013), 1643-1659. doi: 10.1007/s00033-013-0353-x.
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