Summary: We consider the system \[ u_t= \Delta u+v^p,\quad v_t=\Delta v,\quad x\in\mathbb{R}^N_+,\;t>0, \]
\[ -{\partial u\over\partial x_1}=0,\quad -{\partial v\over\partial x_1}= u^q,\quad x_1=0,\;t>0,\quad u(x,0)= u_0(x),\quad v(x,0)= v_0(x),\quad x\in\mathbb{R}^N_+, \] where \(\mathbb{R}^N_+=\{(x_1,x')|x'\in\mathbb{R}^{N-1},x_1>0\}\), \(p,q>0\), and \(u_0\), \(v_0\) are nonnegative and bounded. We prove that if \(pq\leq 1\) every nonnegative solution is global. When \(pq>1\) we let \(\alpha=(p+2)/2(pq-1)\), \(\beta=(2q+1)/2(pq-1)\). We show that if \(\max(\alpha,\beta)>N/2\) or \(\max(\alpha,\beta)= N/2\) and \(p,q\geq 1\), then all nontrivial nonnegative solutions are nonglobal; whereas if \(\max(\alpha,\beta)<N/2\) there exist both global and nonglobal nonnegative solutions.