  We prove that the steady state of a class of multidimensional
reaction-diffusion systems is asymptotically stable at the intersection of
unweighted space and exponentially weighted Sobolev spaces, and pay particular
attention to a special case, namely, systems of equations that arise in
combustion theory. The steady-state solutions considered here are the end
states of the traveling fronts associated with the systems, and thus the
present results complement recent papers \cite{GLS1, GLS2, GLS3, GLSR, GLY}
that study the stability of traveling fronts.
