Catastrophe Theory Concepts for Ignition/Extinction Phenomena

Abstract

This investigation centers on a class of semi-linear elliptic boundary value problems which are to model the physical behaviour of certain stationary reaction-diffusion systems for which ignition and extinction is known to occur. It is shown that the solutions may be represented by the points of a state surface in an appropriate finite-dimensional state space and hence may be classified according to the concepts of Catastrophe Theory via an investigation of the projection image of the state surface in a subspace spanned by the control parameters which form a subset of the state variables. Due to the existence of a distinguished control parameter (which is designated by λ and is known as the “Frank-Kamenetzkiparameter”), the state surface is given by an explicit expression for λ in terms of the cther state variables. With the help of the maximum principle for elliptic differential equations another surface is shown to exist which — in terms of λ — bounds the state surface from below. For this bounding surface, the projection image in the subspace of the control parameters furnishes a bifurcation set which provides lower bounds for the associated critical singularities of the exact problem.