STABILITY OF FRONTS FOR A KPP-SYSTEM, II - THE CRITICAL CASE

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.