Stability of Fronts for a KPP-System, II: The Critical Case

We study a system ofn+1 coupled semilinear parabolic equations on the real line, which depends on a small parameterλand reduces to the scalar Kolmogorov–Petrovsky–Piscounov (KPP) equation, whenλ=0. Under appropriate scaling, the system has a family of traveling fronts, parametrized by their speedγ, when γ 2, as in the scalar KPP case. The case of critical speed,γ=−2 say, is investigated and it is shown that the system inherits some crucial properties of the KPP equation, whenλis small: in particular, the asymptotic stability of the front in a local and semiglobal sense. First, we describe the properties of the front and then apply functional arguments to prove its local stability in an adequate weighted 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 equation.