Global existence and blowup solutions for the Gierer-Meinhardt system (Q1988404)

This article discusses the study of the Gierer-Meinhardt system in the form \[ \begin{cases} u_t=\Delta u-u+\frac{u^p}{v^q} &\quad x\in \Omega, t>0\\ v_t=\Delta v-\frac{v}{R}+\frac{u^r}{Rv^s} &\quad x\in \Omega, t>0\\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0&\quad x\in \partial \Omega, t>0\\ u(x,0)=u_0(x)>0, v(x,0)=v_0(x)>0 &\quad x\in \Omega, \end{cases} \] where \(\Omega\) is asmooth and bounded domain in \({\mathbb R}^N\), \(p>1\), \(q,r,s>0\). There are three main results in the article which present new sufficient conditions for global existence and finite time blow-up of solutions.