On the Gierer-Meinhardt system with precursors

Juncheng Wei; Matthias Winter

doi:10.3934/dcds.2009.25.363

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2009, Volume 25, Issue 1: 363-398. Doi: 10.3934/dcds.2009.25.363

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On the Gierer-Meinhardt system with precursors

Juncheng Wei^1, and
Matthias Winter^2,

1.
Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
2.
Brunel University, Department of Mathematical Sciences, Uxbridge UB8 3PH

Received: April 30, 2007

Revised: December 31, 2007

Published: December 2008

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Abstract

We consider the following Gierer-Meinhardt system with a precursor $ \mu (x)$ for the activator $A$ in $\mathbb{R}^1$:

$A_t=$ε²$A^{''}- \mu (x) A+\frac{A^2}{H} \mbox{ in } (-1, 1),$
$\tau H_t=D H^{''}-H+ A^2 \mbox{ in } (-1, 1),$
$ A' (-1)= A' (1)= H' (-1) = H' (1) =0.$

Such an equation exhibits a typical Turing bifurcation of the second kind, i.e., homogeneous uniform steady states do not exist in the system.
We establish the existence and stability of $N-$peaked steady-states in terms of the precursor $\mu(x)$ and the diffusion coefficient $D$. It is shown that $\mu (x)$ plays an essential role for both existence and stability of spiky patterns. In particular, we show that precursors can give rise to instability. This is a new effect which is not present in the homogeneous case.

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Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J55, 92C15, 92C40.

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