Blow-up for a parabolic system coupled in an equation and a boundary condition (Q2784202)

The paper deals with non-negative solutions of the system NEWLINE\[NEWLINE \begin{aligned} & u_t=\Delta u+v^p,\quad v_t=\Delta v\quad \text{in} \Omega\times (0,\infty),\\ & \partial u/\partial\nu=0,\quad \partial v/\partial\nu=u^q \quad \text{on} \partial\Omega\times (0,\infty),\\ & u(x,0)=u_0(x),\quad v(x,0)=v_0(x)\quad \text{in} \Omega.\end{aligned}NEWLINE\]NEWLINE It is proved that if \(pq\leq 1,\) then every solution is global while if \(pq>1,\) all solutions blow up in finite time. Moreover, if \(p,\;q\geq 1\) then blow-up can occur only on the boundary.