Taiwanese Journal of Mathematics

Stability of Traveling Wave Fronts for Nonlocal Diffusion Equation with Delayed Nonlocal Response

Hongmei Cheng and Rong Yuan

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In this paper, we consider with the stability of traveling wave fronts for the nonlocal diffusion equation with delay and global response. We first establish the existence and comparison theorem of solutions for the nonlocal reaction-diffusion equation by appealing to the theory of abstract functional differential equation. Then we further show that the traveling wave fronts are asymptotical stability with phase shift. Our main technique is the super and subsolution method coupled with the comparison principle and squeezing method.

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Taiwanese J. Math., Volume 20, Number 4 (2016), 801-822.

First available in Project Euclid: 1 July 2017

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Zentralblatt MATH identifier

Primary: 35R10: Partial functional-differential equations 35B40: Asymptotic behavior of solutions 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx] 58D25: Equations in function spaces; evolution equations [See also 34Gxx, 35K90, 35L90, 35R15, 37Lxx, 47Jxx]

nonlocal diffusion asymptotic stability traveling wave fronts super and subsolution comparison principle squeezing method delayed nonlocal response


Cheng, Hongmei; Yuan, Rong. Stability of Traveling Wave Fronts for Nonlocal Diffusion Equation with Delayed Nonlocal Response. Taiwanese J. Math. 20 (2016), no. 4, 801--822. doi:10.11650/tjm.20.2016.6284. https://projecteuclid.org/euclid.twjm/1498874492

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  • J. Al-Omari and S. A. Gourley, Monotone travelling fronts in age-structured reaction-diffusion model of a single species, J. Math. Biol. 45 (2002), no. 4, 294–312.
  • D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, 5–49; Lecture Notes in Math. 446, Springer, Berlin, 1975.
  • P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal. 138 (1997), no. 2, 105–136.
  • N. F. Britton, Reaction-diffusion Equations and Their Applications to Biology, Academic Press, Lodon, 1986.
  • ––––, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math. 50 (1990), no. 6, 1663–1688.
  • J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2433–2439.
  • X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2 (1997), no. 1, 125–160.
  • X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002), no. 2, 549–569.
  • J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl. 185 (2006), no. 3, 461–485.
  • J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations 244 (2008), no. 12, 3080–3118.
  • J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal. 60 (2005), no. 5, 797–819.
  • ––––, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 4, 727–755.
  • D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes Mathemathics Series 279, Longman Scientific & Technical, Harlow, 1992.
  • P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag, Berlin, 1979.
  • P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal. 65 (1977), no. 4, 335–361.
  • R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics 7 (1937), no. 4, 355–369.
  • ––––, The Genetical Theory of Natural Selection: A Complete Variorum Edtion, Oxford University Press, 1999.
  • V. Gubernov, G. N. Mercer, H. S. Sidhu and R. O. Weber, Evans function stability of combustion waves, SIAM J. Appl. Math. 63 (2003), no. 4, 1259–1275.
  • S. Guo and J. Zimmer, Travelling wavefronts in discrete reaction-diffusion equations with nonlocal delay effects, arXiv:1406.5321, 2014.
  • ––––, Stability of travelling wavefronts in discrete reaction-diffusion equations with nonlocal delay effects, Nonlinearity, 28 (2015), no. 2, 463–492.
  • H. Ikeda, Y. Nishiura and H. Suzuki, Stability of traveling waves and a relation between the Evans function and the SLEP equation, J. Reine Angew. Math. 475 (1996), 1–37.
  • R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc. 321 (1990), no. 1, 1–44.
  • ––––, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math. 413 (1991), 1–35.
  • J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci. 184 (2003), no. 2, 201–222.
  • 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), no. 6, 2762–2790. Erratum: Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal. 44 (2012), no. 1, 538–540. http://dx.doi.org/10.1137/110850633
  • J. D. Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications, Third edition, Interdisciplinary Applied Mathematics 18, Springer-Verlag, New York, 2003.
  • S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl. 346 (2008), no. 2, 415–424.
  • S. Pan, W.-T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys. 60 (2009), no. 3, 377–392.
  • ––––, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal. 72 (2010), no. 6, 3150–3158.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983.
  • S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 5, 991–1011.
  • D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355.
  • K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amr. Math. Soc. 302 (1987), no. 2, 587–615.
  • H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal. 31 (2000), no. 3, 514–534.
  • J. W.-H. So and X. Zou, Traveling waves for the diffusive Nicholson's blowflies equation, Appl. Math. Comput. 122 (2001), no. 3, 385–392.
  • A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs 140, American Mathematical Society, Providence, RI, 1994.
  • 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), no. 1, 153–200.
  • ––––, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations 20 (2008), no. 3, 573–607.
  • Z.-X. Yu and R. Yuan, Travelling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM J. 51 (2009), no. 1, 49–66.
  • ––––, Existence, asymptotic and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response, Taiwanese J. Math. 17 (2013), no. 6, 2163–2190.
  • L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. Differential Equations 197 (2004), no. 1, 162–196.