Topological Techniques in Reaction-Diffusion Equations

Abstract

In this note, we shall illustrate how some topological ideas can be used to obtain rather precise information about solutions of reaction-diffusion equations. The equations are of the form

$$ {{\text{u}}_t} = {u_{{xx}}} + {\text{f}}(u), - {\text{L}} < x < {\text{L}} $$

((1))

in a single space variable, with either homogeneous Dirichlet or Neumann boundary conditions. The function f(-∞) is a cubic polynomial having three distinct real roots, with f(-∞) > 0 > f(∞). The solutions of interest to us are the so-called “steady-state” solutions; i.e. solutions which are independent of t, and therefore satisfy the equation

$$ {\text{u''}}\,{\text{ + f}}(u) = 0, - {\text{L}} < x < {\text{L}} $$

((2))

with the same boundary conditions. Our first goal is to determine the dimension of the unstable manifold of solutions of (2); the solution being considered as a particular solution of (1). Our second goal is to show how to “continue” solutions when the boundary manifolds serve as the “continuation parameter.”

Researchof Conley supported by N.S.F., that of Smoller by A.F.O.S.R.