Rigorous Asymptotic Expansions for Critical Wave Speeds in a Family of Scalar Reaction-Diffusion Equations

被引：0
作者：
Nikola Popović
Tasso J. Kaper
机构：
[1] Boston University,Center for BioDynamics and Department of Mathematics and Statistics
来源：
Journal of Dynamics and Differential Equations | 2006年 / 18卷
关键词：
Reaction-diffusion equations; FKPP equation; traveling waves; critical wave speeds; geometric desingularization; blow-up technique; 35K57; 34E15; 34E05;
D O I：
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摘要：
We investigate traveling wave solutions in a family of reaction-diffusion equations which includes the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with quadratic nonlinearity and a bistable equation with degenerate cubic nonlinearity. It is known that, for each equation in this family, there is a critical wave speed which separates waves of exponential decay from those of algebraic decay at one of the end states. We derive rigorous asymptotic expansions for these critical speeds by perturbing off the classical FKPP and bistable cases. Our approach uses geometric singular perturbation theory and the blow-up technique, as well as a variant of the Melnikov method, and confirms the results previously obtained through asymptotic analysis in [J.H. Merkin and D.J. Needham, (1993). J. Appl. Math. Phys. (ZAMP) A, vol. 44, No. 4, 707–721] and [T.P. Witelski, K. Ono, and T.J. Kaper, (2001). Appl. Math. Lett., vol. 14, No. 1, 65–73].
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共 32 条
[11]  
Fenichel N.(2001)Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions SIAM J. Math. Anal. 33 286-314
[12]  
Fenichel N.(1993)Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension J. Appl. Math. Phys. (ZAMP) A 44 707-721
[13]  
Fisher R.A.(1999)Reaction-diffusion and phase waves occurring in a class of scalar reaction-diffusion equations Nonlinearity 12 41-58
[14]  
Guckenheimer J.(1983)Sustained resonance for a nonlinear system with slowly varying coefficients SIAM J. Math. Anal. 14 847-860
[15]  
Hoffman K.(1987)The Mel’nikov technique for highly dissipative systems SIAM J. Appl. Math. 47 232-243
[16]  
Weckesser W.(1996)Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation IMA J. Appl. Math. 56 289-302
[17]  
Kolmogorov A.N.(2001)Critical wave speeds for a family of scalar reaction-diffusion equations Appl. Math. Lett. 14 65-73
[18]  
Petrowskii I.G.(undefined)undefined undefined undefined undefined-undefined
[19]  
Piscounov N.(undefined)undefined undefined undefined undefined-undefined
[20]  
Krupa M.(undefined)undefined undefined undefined undefined-undefined
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