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Title:  On the GiererMeinhardt System with Saturation 
Authors:  Winter, M Wei, J 
Keywords:  Saturation, Nonlocal Eigenvalue Problem, Stability 
Publication Date:  2004 
Publisher:  World Scientific 
Citation:  Comm Contemp Math 6 (2004), 403430 
Abstract:  We consider the following shadow GiererMeinhardt system with saturation:
\left\{\begin{array}{l}
A_t=\epsilon^2 \Delta A A + \frac{A^2}{ \xi (1+k A^2)} \ \ \mbox{in} \ \Omega \times (0, \infty),\\
\tau \xi_t= \xi +\frac{1}{\Omega} \int_\Om A^2\,dx
\ \ \mbox{in} \ (0, +\infty),
\frac{\partial A}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega\times(0,\infty),
\end{array}
\right.
where \ep>0 is a small parameter, $\tau \geq 0,\, k>0
and \Omega \subset R^n is smooth bounded domain.
The case k=0 has been studied by many authors in recent years.
Here we give some sufficient conditions on $k$ for the existence and stability of stable spiky solutions.
In the onedimensional case we have a complete answer to the stability behavior.
Central to our study are a parameterized groundstate equation
and the associated nonlocal eigenvalue problem (NLEP)
which is solved by functional analysis arguments and the continuation method. 
URI:  http://bura.brunel.ac.uk/handle/2438/570 
Appears in Collections:  Mathematical Science Dept of Mathematics Research Papers

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