The asymptotic critical wave speed in a family of scalar reaction-diffusion equations

Journal of Mathematical Analysis and Applications

Volumn 326, Issue 2, 2007, Pages 1007-1023

  Dumortier, Freddy ^a   Popović, Nikola ^b   Kaper, Tasso J ^b  

^a HASSELT UNIVERSITY   (Belgium)

^b Boston University   (United States)

Author keywords

Asymptotic expansions; Blow up technique; Critical wave speeds; Reaction diffusion equations; Traveling waves

Indexed keywords

EID: 33750685310     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2006.03.050     Document Type: Article

References (27)

1. Billingham J. Phase plane analysis of one-dimensional reaction-diffusion waves with degenerate reaction terms. Dyn. Stab. Syst. 15 1 (2000) 23-33
2. Britton N.F. Reaction-Diffusion Equations and Their Applications to Biology (1986), Academic Press Inc., London
3. Dumortier F. Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations. In: Schlomiuk D. (Ed). Bifurcations and Periodic Orbits of Vector Fields. NATO Sci. Ser. C Math. Phys. Sci. vol. 408 (1993), Kluwer Acad. Publ., Dordrecht 19-73
4. Dumortier F., and De Maesschalck P. Topics in singularities and bifurcations of vector fields. In: Ilyashenko Y., Rousseau C., and Sabidussi G. (Eds). Normal Forms, Bifurcations, and Finiteness Problems in Differential Equations. NATO Sci. Ser. II Math. Phys. Chem. vol. 137 (2004), Kluwer Acad. Publ., Dordrecht 33-86
5. Dumortier F., and Roussarie R. Canard cycles and center manifolds. Mem. Amer. Math. Soc. 121 577 (1996)
6. Dumortier F., and Roussarie R. Geometric singular perturbation theory beyond normal hyperbolicity. In: Jones C.K.R.T., and Khibnik A. (Eds). Multiple-time-scale dynamical systems. IMA Vol. Math. Appl. vol. 122 (2001), Springer-Verlag, New York 29-63
7. Dumortier F., Roussarie R., and Sotomayor J. Bifurcations of cuspidal loops. Nonlinearity 10 6 (1997) 1369-1408
8. Fenichel N. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21 (1971) 193-226
9. Fenichel N. Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations 31 1 (1979) 53-98
10. Fisher R.A. The wave of advance of advantageous genes. Ann. Eugenics 7 (1937) 355-369
11. Jones C.K.R.T. Geometric singular perturbation theory. Dynamical Systems. Lecture Notes in Math. vol. 1609 (1995), Springer-Verlag, New York 44-118
12. Kolmogorov A.N., Petrowskii I.G., and Piscounov N. Etude de l'équation de la diffusion avec croissance de la quantité de matiére et son application à un problème biologique. Moscow Univ. Math. Bull. 1 (1937) 1-25
13. Krupa M., and Szmolyan P. Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions. SIAM J. Math. Anal. 33 2 (2001) 286-314
14. Krupa M., and Szmolyan P. Relaxation oscillation and canard explosion. J. Differential Equations 174 2 (2001) 312-368
15. Leach J.A., Needham D.J., and Kay A.L. The evolution of reaction-diffusion waves in a class of reaction-diffusion equations: Algebraic decay rates. Phys. D 167 3-4 (2002) 153-182
16. P. De Maesschalck, Gevrey properties of real, planar singularly perturbed systems, preprint, 2005
17. Merkin J.H., and Needham D.J. Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension. J. Appl. Math. Phys. (ZAMP) A 44 4 (1993) 707-721
18. Needham D.J., and Barnes A.N. Reaction-diffusion and phase waves occurring in a class of scalar reaction-diffusion equations. Nonlinearity 12 1 (1999) 41-58
19. K. Ono, Analytical methods for reaction-diffusion equations: Critical wave speeds and axi-symmetric phenomena, PhD thesis, Boston University, Boston, MA, USA, 2001
20. Popović N. A geometric analysis of logarithmic switchback phenomena. In: Mortell M.P., O'Malley Jr. R.E., Pokrovskii A.V., and Sobolev V.A. (Eds). International Workshop on Hysteresis & Multi-scale Asymptotics, University College Cork. Ireland, 17-21 March 2004. J. Phys. Conf. Ser. vol. 22 (2005), Inst. Phys. Publishing, Bristol 164-173
21. Popović N., and Kaper T.J. Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction-diffusion equations. J. Dynam. Differential Equations 18 1 (2006) 77-114
22. Popović N., and Szmolyan P. A geometric analysis of the Lagerstrom model problem. J. Differential Equations 199 2 (2004) 290-325
23. Popović N., and Szmolyan P. Rigorous asymptotic expansions for Lagerstrom's model equation-A geometric approach. Nonlinear Anal. 59 4 (2004) 531-565
24. Ramis J.P. Séries divergentes et théories asymptotiques. Bull. Soc. Math. France 121 (1993)
25. Sherratt J.A., and Marchant B.P. Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation. IMA J. Appl. Math. 56 3 (1996) 289-302
26. van Gils S., Krupa M., and Szmolyan P. Asymptotic expansions using blow-up. Z. Angew. Math. Phys. 56 3 (2005) 369-397
27. Witelski T.P., Ono K., and Kaper T.J. Critical wave speeds for a family of scalar reaction-diffusion equations. Appl. Math. Lett. 14 1 (2001) 65-73