Milti-Bump Ground States of the Gierer-Meinhardt System in ${ I\!\!R}^2$

Manuel del Pino
Departamento de Ingeniería Matemática
and Centro de Modelamiento Matemático Universidad de Chile,
Casilla 170 Correo 3, Santiago, CHILE
email: delpino@dim.uchile.cl


Micha\l Kowalczyk
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, U.S.A.
email: kowalcyk@asdf1.math.cmu.edu


Juncheng Wei
Department of Mathematics
The Chinese University of Hong Kong
Shatin, Hong Kong
email: wei@math.cuhk.edu.hk

ABSTRACT:

We consider the stationary Gierer-Meinhardt system in ${ I\!\!R}^2$:

\begin{displaymath}\left\{ \begin{array}{ll} \Delta A- A+\frac{A^2}{H}=0 &\mbox... ...o 0 \ \mbox{as} \ \vert x\vert \to +\infty. \end{array}\right. \end{displaymath}

We construct multi-bump ground-state solutions for this system for all sufficiently small $\sigma$. The centers of these bumps are located at the vertices of a regular polygon, and they resemble after a suitable scaling in their A-coordinate, the unique radially symmetric solution of

\begin{displaymath}\left\{ \begin{array}{ll} \Delta w - w + w^2=0 \quad \hbox{ ... ...0 \quad \hbox{ as } \vert y\vert\to \infty. \end{array}\right. \end{displaymath}

A similar construction is made for vertices of two concentric polygons, and a general procedure for detection of organized finite patterns is suggested.


Get the paper in its entirety as