  We study the planar front solution for a class of reaction diffusion
equations in multidimensional space in the case when the essential spectrum of
the linearization in the direction of the front touches the imaginary axis. At
the linear level, the spectrum is stabilized by using an exponential weight.
A-priori estimates for the nonlinear terms of the equation governing the
evolution of the perturbations of the front are obtained when perturbations
belong to the intersection of the exponentially weighted space with the
original space without a weight. These estimates are then used to show that in
the original norm, initially small perturbations to the front remain bounded,
while in the exponentially weighted norm, they algebraically decay in time.
