The Gierer & Meinhardt system:
the breaking of homoclinics
and multi-bump ground states

Manuel del Pino
Departamento de Ingeniería Matemática
Universidad de Chile,
Casilla 170 Correo 3, Santiago, CHILE.
e.mail: delpino@dim.uchile.cl

Micha $\l$ Kowalczyk
Department of Mathematical Sciences
Carnegie Mellon University,
Pittsburgh, PA 15213, U.S.A.
e.mail: kowalcyk@andrew.cmu.edu

Xinfu Chen
Department of Mathematics & Statistics
University of Pittsburgh
Pittsburgh, PA 15260, U.S.A.

ABSTRACT:

In this paper we study ground-states of the Gierer & Meinhardt system on the line, namely solutions of the problem

u'' -u + u²/ v = 0

$\begin{displaymath}\sigma^{-2} v'' - v + u^2 = 0 \end{displaymath}$

$\begin{displaymath}u,v > 0, \quad u(\pm \infty ) = 0= v(\pm \infty ). \end{displaymath}$

We prove that given any number N, there exists a solution to this problem exhibiting exactly N bumps in its u-component, separated from each other at a distance $O(\vert\log\sigma \vert)$ , whenever $\sigma$ is sufficiently small. These bumps resemble the shape of the unique solution of

$\begin{displaymath}U'' -U + U^2 = 0 , \quad 0<U(\pm \infty ) =0, \ U'(0) =0 . \end{displaymath}$

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