Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 84, pp. 1-12.
Title: Constructing universal pattern formation processes
governed by reaction-diffusion systems
Author: Sen-Zhong Huang (Univ. Rostock, Germany)
Abstract:
For a given connected compact subset $K$ in
$\mathbb{R}^n$ we construct a smooth map $F$ on
$\mathbb{R}^{1+n}$ in such a way that the corresponding
reaction-diffusion system
$u_t=D\Delta u+F(u)$ of $n+1$
components $u=(u_0,u_1,\dots ,u_n)$, accompanying with the
homogeneous Neumann boundary condition, has an attractor
which is isomorphic to $K$. This implies the following
universality: The make-up of a pattern with arbitrary complexity
(e.g., a fractal pattern) can be realized by a reaction-diffusion
system once the vector supply term $F$ has been previously
properly constructed.
Submitted August 28, 2002. Published October 4, 2002.
Math Subject Classifications: 35B40, 70G60, 35Q99
Key Words: Attractor; pattern formation.