G. Raugel et K. Kirchgassner, STABILITY OF FRONTS FOR A KPP-SYSTEM, II - THE CRITICAL CASE, Journal of differential equations (Print), 146(2), 1998, pp. 399-456
We study a system of n + 1 coupled semilinear parabolic equations on t
he real line, which depends on a small parameter lambda and reduces to
the scaler Kolmogorov-Petrovsky-Piscounov (KPP) equation, when lambda
= 0. Under appropriate scaling, the system has a family of traveling
fronts, parametrized by their speed gamma, when \gamma\ greater than o
r equal to 2, as in the scalar KPP case. The case of critical speed, g
amma = -2 say, is investigated and it is shown that the system inherit
s some crucial properties of the KPP equation, when l.is small: in par
ticular, the asymptotic stability of the front in a local and semiglob
al sense. First, we describe the properties of the front and then appl
y functional arguments to prove its local stability in an adequate wei
ghted Sobolev space. Moreover, the decay rate of the perturbations is
shown to be polynomial in time. Finally we show a semiglobal stability
property of the front, which also is inherited from the scalar KPP eq
uation. (C) 1998 Academic Press.