In the following we will discuss some known results on the behavior of solutions to reaction-diffusion equations. We will be concerned with the stability of steady-state solutions in different classes of domains. A result in Matano states that for convex domains, every non-constant stable steady-state solution to the reaction-diffusion equation is unstable. As an application of a theorem in Matano, we show that this result for convex domains does not generalize to the larger class of star-shaped domains.

Author Bio

The paper was completed during the Summer 2004 REU program at Tulane University. The theme of the applied math program was the effect of diffusion on the solutions of reaction-diffusion equations. During the program, Xuefeng Wang (Tulane University) taught a crash course in PDE's, and Wei-Ming Ni (University of Minnesota) delivered a series of lectures on reaction-diffusion systems. Thereafter, my group worked with John Alford (Tulane University) studying the solutions of reaction-diffusion equations. We found numerical evidence suggesting the existence of non-constant stable steady-state solutions to a reaction-diffusion equation in two types of star-shaped domains. In addition, we numerically analyzed the relationship between the diffusion constant, the aperture width, and the nature of the steady-state.