Table of Contents

Title Page

Copyright Page

Chapter 7 - Equilibrium Models

7.1 Elliptic Models

7.2 Theoretical Results

7.3 Eigenvalue Problems

7.4 Stability and Bifurcation

eISBN : 978-0-470-22595-0

7

Equilibrium Models

In this final chapter we address some problems in the theory of nonlinear elliptic partial differential equations. We consider only some representative examples, many of which are elementary; we do not lay out a general theory. One way to think about elliptic equations is that they are equilibrium problems born out of the time-asymptotic limit of reaction-diffusion equations. For example, steady-state solutions, if they exist, of the parabolic reaction-diffusion equation must satisfy the equilibrium equation The latter semilinear equation is a steady-state equation in that time does not appear, and = a function of the spatial variable only. So one may think of the solutions of (7.0.1) as evolving into the solutions = of (7.0.2) as gets large. One of the major problems in the study of reaction-diffusion equations is to confirm this observation in special cases, and it leads to questions of existence of solutions of both (7.0.1) and (7.0.2) and also to questions of stability and bifurcation.