Dynamic transitions and bifurcations of 1D reaction-diffusion equations:
The Self-adjoint case
Abstract
This paper deals with the classification of transition phenomena in the
most basic dissipative system possible, namely the 1D reaction diffusion
equation. The emphasis is on the relation between the linear and
nonlinear terms and the effect of the boundaries which influence the
first transitions. We consider the cases where the linear part is
self-adjoint with 2nd order and 4th order derivatives which is the case
which most often arises in applications. We assume that the nonlinear
term depends on the function and its first derivative which is basically
the semilinear case for the second order reaction-diffusion system. As
for the boundary conditions, we consider the typical Dirichlet, Neumann
and periodic boundary settings. In all the cases, the equations admit a
trivial steady state which loses stability at a critical parameter. We
aim to classify all possible transitions and bifurcations that take
place. Our analysis shows that these systems display all three types of
transitions: continuous, jump and mixed and display transcritical,
supercritical bifurcations with bifurcated states such as finite
equilibria, circle of equilibria, and slowly rotating limit cycle. Many
applications found in the literature are basically corollaries of our
main results. We apply our results to classify the first transitions of
the Chaffee-Infante equation, the Fisher-KPP equation, the Kuramoto
Sivashinsky equation and the Swift-Hohenberg equation.