Asymptotics and Uniqueness of Travelling Waves for Non-Monotone Delayed Systems on 2D Lattices

Yu, Zhi-Xian ; Mei, Ming

Canadian mathematical bulletin, Tome 56 (2013) no. 3, pp. 659-672

Résumé

We establish asymptotics and uniqueness (up to translation) of travelling waves for delayed $2\text{D}$ lattice equations with non-monotone birth functions. First, with the help of Ikehara’s Theorem, the a priori asymptotic behavior of travelling wave is exactly derived. Then, based on the obtained asymptotic behavior, the uniqueness of the traveling waves is proved. These results complement earlier results in the literature.

Détail

DOI : 10.4153/CMB-2011-180-4

Mots-clés : 35K57, 2D lattice systems, traveling waves, asymptotic behavior, uniqueness, nonmonotone nonlinearity

Bibliographie

[1] [1] Aguerrea, M., Trofimchuk, S., and Valenzuela, G., Uniqueness of fast travelling fronts in reaction-diffusion equations with delay. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464 (2008, 2591–2608. Google Scholar | DOI

[2] [2] Berestyski, H. and Nirenberg, L., Traveling fronts in cylinders. Ann. Inst. H. Poincaré Anal Non. Linéaire 9 (1992, no. 5, 497–572. Google Scholar

[3] [3] Chan, J.W., Mallet-Paret, J., and Vleck, E. S., Travelling wave solutions for systems of ODEs on a two-dimensional spatial lattice. SIAM J. Appl. Math. 59 (2006, no. 2, 455–493. Google Scholar

[4] [4] Carr, J. and Chmaj, A., Uniqueness of travelling waves for nonlocal monostable equations. Proc. Amer. Math. Soc. 132 (2004, 2433–2439. Google Scholar | DOI

[5] [5] Chen, X., Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differential Equations 2 (1997, no. 1, 125–160. Google Scholar

[6] [6] Chen, X., S.-Fu, C., and J.-Guo, S., Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices. SIAM J. Math. Anal. 38 (2006, no. 1, 233–258. Google Scholar | DOI

[7] [7] Chen, X. and J.-Guo, S., Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math. Ann. 326 (2003, no. 1, 123–146. Google Scholar | DOI

[8] [8] Cheng, C., Li, W., and Wang, Z., Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice. IMA J. Appl. Math. 73 (2008, no. 4, 592–618. Google Scholar | DOI

[9] [9] Cheng, C., Asymptotic stability of traveling wavefronts in a delayed population model stage structure on a two-dimensional spatial lattice. Discrete Contin. Dyn. Syst. Ser. B 13 (2010, no. 3, 559–575. Google Scholar | DOI

[10] [10] Coville, J., On uniqueness and monotonicity of solutions of nonlocal reaction diffusion equation. Ann. Mat. Pura Appl. 185 (2006, no. 3, 461–485. Google Scholar | DOI

[11] [11] Diekmann, O. and Kaper, H. G., On the bounded solutions of a nonlinear convolution equation. Nonlinear Anal. 2 (1978, no. 6, 721–737. Google Scholar | DOI

[12] [12] Fang, J., Wei, J., and Zhao, X.-Q., Spreading speeds and travelling waves for non-monotone time-delayed lattice equations. Proc. R. Soc. A Math. Phys. Eng. Sci. 466 (2010, no. 2119, 1919–1934. Google Scholar | DOI

[13] [13] Fang, J., Uniqueness of traveling waves for nonlocal lattice equations. Proc. Amer. Math. Soc. 139 (2011, no. 4, 1361–1373. Google Scholar | DOI

[14] [14] Faria, T. and Trofimchuk, S., Nonmonotone travelling waves in a single species reaction-diffusion equation with delay. J. Differential Equations 228 (2006, no. 1, 357–376. Google Scholar | DOI

[15] [15] Ma, S., Traveling waves for non-local delayed diffusion equations via auxiliary equations. J. Differential Equations 237 (2007, no.2, 259–277. Google Scholar | DOI

[16] [16] Ma, S. and Wu, J., Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation. J. Dynam. Differential Equations 19 (2007, no. 2, 391–436. Google Scholar | DOI

[17] [17] Ma, S., Weng, P., and Zou, X., Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation. Nonlinear Anal. 65 (2006, no. 10, 1858–1890. Google Scholar | DOI

[18] [18] Ma, S. and Zou, X., Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equations with delay. J. Differential Equations 217 (2005, no. 1, 54–87. Google Scholar | DOI

[19] [19] Weng, P., Huang, H., and Wu, J., Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction. IMA J. Appl. Math. 68 (2003, no. 4, 409–439. Google Scholar | DOI

[20] [20] Widder, D. V., The Laplace Tranform. Princeton Mathematical Series, 6, Princeton University Press, Princeton, NJ, 1941. Google Scholar