October  2006, 14(4): 737-751. doi: 10.3934/dcds.2006.14.737

Global existence of solutions of an activator-inhibitor system

1. 

School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455, United States

Received  April 2005 Revised  August 2005 Published  January 2006

We consider the generalized Gierer-Meinhardt system

$\frac{\partial u_{j}}{\partial t}=d_{j}$Δ $u_{j}- a_{j}u_{j} +g_{j}(x,u) \ \ \text{in}\ \ \Omega\times[0,T) $ ,
$\frac{\partial u_{j}}{\partial\nu}=0\ \ \text{on}\ \ \partial\Omega\times[ 0,T) $,
$u_{j}(x,0) =\varphi_{j}(x)\ \ \text{in}\ \ \Omega$

where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{n}$ with $\nu$ its unit outer normal, $j=1,2$, $u=(u_1,u_2)$ and

$g_{1}(x,u) =\rho_{1}(x,u) \frac{u_{1}^{p}}{u_{2}^{q}}+\sigma_{1}(x) $ ,
$g_{2}(x,u) =\rho_{2}(x,u) \frac{u_{1}^{r}}{u_{2}^{s}}+\sigma_{2}(x) $.

Here $d_{j}, a_{j}$ are positive constants, $\rho_{1}\geq0$, $\rho _{2}>0,\sigma_{j}\geq0$ are bounded smooth functions and $p,q,r,s$ are nonnegative constants satisfying $0<\frac{p-1}{r}<\frac{q}{s+1}$.
We show that there is a unique global solution when p-1 < r, which improves 1987 result of K. Masuda and K. Takahashi [10]. Asymptotic bounds of global solutions are also established which yield new a priori estimates of stationary solutions.

Citation: Huiqiang Jiang. Global existence of solutions of an activator-inhibitor system. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 737-751. doi: 10.3934/dcds.2006.14.737
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