Focant S., Gallay Th.
Existence and Stability of Propagating Fronts
for an Autocatalytic Reaction-Diffusion System
(122K, (uuencoded gzipped) Postscript)
ABSTRACT. We study a one-dimensional reaction-diffusion system which describes
an isothermal autocatalytic chemical reaction involving both a
quadratic (A + B -> 2B) and a cubic (A + 2B -> 3B) autocatalysis.
The parameters of this system are the ratio D = D_B/D_A of the
diffusion constants of the reactant A and the autocatalyst B,
and the relative activity k of the cubic reaction. First, for all
values of D > 0 and k >= 0, we prove the existence of a family
of propagating fronts (or travelling waves) describing the advance
of the reaction. In particular, in the quadratic case k=0, we recover
the results of Billingham and Needham [BN]. Then, if D is close to
1 and k is sufficiently small, we prove using energy functionals that
these propagating fronts are stable against small perturbations in
exponentially weighted Sobolev spaces. This extends to our system part
of the stability results which are known for the scalar Fisher equation.