STABILITY OF WEAK SHOCKS IN LAMBDA OMEGA SYSTEMS

We consider the lambda - omega reaction diffusion equation. Howard and Kopell ([HK]) show that these equations support the existence of "weak shocks," which are travelling wave solutions that join plane waves having sufficiently close wave numbers. Furthermore, they show that the wave satisfies an entropy condition at the end states. In studying the stability of these solutions we are faced with the difficulty that in an unweighted space the essential spectrum contains gamma = 0. However, in an exponentially weighted L infinity space the spectrum is contained in the left-half-plane. We show that these solutions are stable in a polynomially weighted L infinity space. Furthermore, we show that the rate at which the perturbation decays as t --> infinity depends on the growth rate of the polynomial near infinity.