Electron. J. Diff. Eqns., Vol. 2002(2002), No. 84, pp. 1-12.

Constructing universal pattern formation processes governed by reaction-diffusion systems

Sen-Zhong Huang

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.

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Sen-Zhong Huang
Fachbereich Mathematik
Universitat Rostock
Universitatsplatz 1
D-18055 Rostock, F. R. Germany
e-mail: sen-zhong.huang@mathematik.uni-rostock.de

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