Topological techniques in reaction-diffusion equations

Abstract

We consider the 'stability' problem for steady-state solutions of reaction-diffusion equations of the form u, = u, + f(u), -L 0 for u 0 for u > B, B > A. Using isolating block techniques, together with some new estimates for elliptic functions, we are able to determine the number of such solutions, and the dimension of their unstable manifolds. In particular we show that non-constant steady-state solutions with homogeneous Neumann boundary conditions cannot be stable, but for homogeneous Dirichlet data, stable non-constant solutions can exist. These questions are connected with the problem of 'secondary' bifurcation for such solutions.