Critical exponents in a parabolic system with inner absorption and coupled nonlinear boundary flux (Q1883212)

The following parabolic system with inner absorption and coupled nonlinear boundary flux is studied: \[ u_t=\Delta u + u^{\alpha_1}, \quad v_t=\Delta v-v^{\beta_1}, \quad x \in \Omega,\;t \in (0,T), \] \[ \frac{\partial u}{\partial \eta}=u^{\alpha_2}v^p, \quad \frac{\partial v}{\partial \eta}=u^q v^{\beta_2}, \quad x\in \partial \Omega,\;t\in(0,T), \] \[ u(x,0)=u_0(x), \quad v(x,0)=v_0(x), \quad x\in {\overline{\Omega}}, \] where \(\Omega \subset \mathbb R^N\) is a bounded domain with smooth boundary \(\partial \Omega\), \(p,q>0\), \(\alpha_i, \beta_i \geq 0\), \(u_0\) and \(v_0\) are functions satisfying the compatibility conditions on \(\overline{\Omega}. \) The main results are conditions of existence of blow up in finite time and existence of global solution. The authors show that blow up and global solvability of the solution depend from numbers \(\tau_1=(p-\beta_2+\gamma)/\gamma_1\), \(\tau_2=(q-\alpha_2+1)/\gamma_2\) for \(\alpha_1 \leq 1,\) where \(\gamma=\max(\frac{\beta_1+1}{2},1), \quad \gamma=pq-(1-\alpha_2)(1-\beta_2).\) The conditions of blow up are \(\alpha_1>1\) for large initial data or \(\tau_1,\tau_2>0, \alpha_2 \leq 1\) for solution large initial data. The solution of the problem is a global if \(\tau_1,\tau_2 \leq 0,\) with \(\alpha_2 \leq 1.\)