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Travelling wave solutions of the degenerate Kolmogorov–Petrovski–Piskunov equation

Published online by Cambridge University Press:  01 July 2003

G. S. MEDVEDEV ,
K. ONO  and
P. J. HOLMES
G. S. MEDVEDEV
Affiliation:
Program in Applied and Computational Mathematics, Princeton University, Einstein Drive, Princeton, NJ 08540, USA email: medvedev@princeton.edu School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
K. ONO
Affiliation:
Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA email: ono@math.bu.edu
P. J. HOLMES
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544-1000, USA email: pholmes@rimbaud.Princeton.edu
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Abstract

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We prove the existence of a family of Travelling Wave (TW) solutions for a large class of scalar reaction-diffusion equations with degenerate, nonlinear diffusion coefficients and monostable nonlinear reaction terms. We also investigate stability. Specifically, we show that, as in the linear diffusion case [6], the slowest TW in the family yields the asymptotic rate of the propagation of disturbances from the unstable rest state in these systems. In addition, we give conditions on the reaction term and diffusion coefficient ensuring the existence of interfaces.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 14 , Issue 3 , June 2003 , pp. 343 - 367
Copyright
2003 Cambridge University Press