Global existence of solutions of an activator-inhibitor 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.