Numerical bifurcation and stability analysis for steady-states of reaction diffusion equations

Abstract

Solutions $u \in H_0^1 (\Omega ) \cap H^2 (\Omega )$ of a semilinear elliptic boundary value problem, $Au + f(x,u,\lambda ) = 0$ (with $f_u (x,u,\lambda )$ bounded below) can be put into a one-to-one correspondence with zeros $c \in \mathbb{R}^d$ of a function $c \to B(c,\lambda ) \in \mathbb{R}^d$. Often d is small. The function $B(c,\lambda )$ is called the bifurcation function. It can also be shown that the eigenvalues of the matrix $B_c (c,\lambda )$ characterize the stability properties of the solutions of the elliptic problem as rest points of $u_t + Au + f(x,u,\lambda ) = 0$. A finite element method that can be used for computing B and B _c has recently been proposed. An overview of these results and the finite element method is given.