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Title: The critical wave speed for the Fisher-Kolmogorov-Petrowskii-Piscounov equation with cut-off
Authors: DUMORTIER, Freddy
Popovic, Nikola
Kaper, Tasso J.
Issue Date: 2007
Citation: NONLINEARITY, 20(4). p. 855-877
Abstract: The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than epsilon = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold e, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c(crit)(epsilon) = 2 - pi(2)/(ln epsilon)(2) + O((ln epsilon)(-3)), as epsilon -> 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of ccrit on e as well as for the universality of the corresponding leading-order coefficient (pi(2)).
Notes: Univ Hasselt, B-3590 Diepenbeek, Belgium. Boston Univ, Ctr Biodynam, Boston, MA 02215 USA. Boston Univ, Dept Math & Stat, Boston, MA 02215 USA.DUMORTIER, F, Univ Hasselt, Campus Diepenbeek,Agoralaan Gebouw D, B-3590 Diepenbeek, Belgium.freddy.dumortier@uhasselt.be popovic@math.bu.edu tasso@math.bu.edu
URI: http://hdl.handle.net/1942/4041
DOI: 10.1088/0951-7715/20/4/004
ISI #: 000245555000004
ISSN: 0951-7715
Category: A1
Type: Journal Contribution
Validation: ecoom, 2008
Appears in Collections: Research publications

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