A classification of spikes and plateaus

In this paper we classify local maxima into spikes and plateaus. We give analytic; definitions for spikes and plateaus in terms of a nonlocal gradient and a fourth order derivative. In higher dimensions the Hesse matrix of Delta f(x) is of relevance. This classification is applied to pattern formation models in mathematical physics and mathematical biology, including Cahn-Hilliard equations, chernotaxis equations, reaction-diffusion equations, GiererMeinhardt models, and Gray-Scott models. We show for some of these examples that the stability of spatial patterns depends on the spike versus plateau type of the solution. We prove, for example, that scalar react ion-d iffusion equations in any spatial dimension cannot have stable spike steady states.