Global asymptotic stability of minimal fronts in monostable lattice equations

Discrete and Continuous Dynamical Systems

Volumn 21, Issue 1, 2008, Pages 259-275

  Ma, Shiwang ^a   Zhao, Xiao Qiang ^b  

^a NANKAI UNIVERSITY   (China)

^b MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

Author keywords

Asymptotic stability; Lattice equations; Minimal speed; Traveling waves

Indexed keywords

EID: 44449164566     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2008.21.259     Document Type: Conference Paper

References (25)

1. D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" (eds., J. A. Goldstein), Springer-Verlag, (1975), 5-49.
2. D. G. Aronson and H. F. Weinberger, Multidimentional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.
3. M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 285 (1983).
4. J. Bricmont and A. Kupiainen, Stability of moving fronts in the Ginzburg-Landau equation, Commun. Math. Phys., 159 (1994), 287-318.
5. X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
6. X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.
7. X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.
8. X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.
9. P. C. Fife and S. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.
10. R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 335-369.
11. T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764.
12. X. Hou and Y. Li, Local stability of traveling wave solutions of nonlinear reaction-diffusion equations, Discrete and Continuous Dynamical Systems, 15 (2006), 681-701.
13. K. Kirchgassner, On the nonlinear dynamics of traveling fronts, J. Differential Equations, 96 (1992), 256-278.
14. A. N. Kolmogorov, I. G. Petrowsky and N. S. Piscounov, Etude de lequation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Univ. dEtat a Moscou, Ser. Intern., A1 (1937), 1-26.
15. X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Communications on Pure and Appl. Math., 60 (2007), 1-40.
16. M. Lucia, C. B. Muratov and M. Novaga, Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium, Comm. Pure Appl. Math., 57 (2004), 616-636.
17. S. Ma and X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.
18. S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.
19. F. Rothe, Convergence to traveling fronts in semilinear parabolic equations, Proc. Roy. Soc. Edinburgh, 80A (1978), 213-234.
20. F. Rothe, Convergence to pushed fronts, Rocky Mountain J. Math., 11 (1981), 617-633.
21. D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
22. D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Differential Equations, 25 (1977), 130-144.
23. H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2001), 514-534.
24. K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.
25. H. F. Weinberger, Long-time behavior of a class of biological models, SIAM, J. Math. Anal., 13 (1982), 353-396.