Stable patterns and Morse index one solutions

Abstract

We survey results on shapes of the stable steady states of two nonlinear problems: a variational problem with a mass constraint and the shadow system of activator-inhibitor type. We see that the stable steady states of the two problems are the Morse index one solutions of a scalar reaction-diffusion equation. We study shapes of the Morse index one solutions and see that the shapes of the Morse index one solutions are deeply related to the "hot spots" conjecture of J. Rauch. We also survey results on the "hot spots" conjecture and related problems.