Blow-up analysis for a system of heat equations coupled through a nonlinear boundary condition (Q5938898)

The authors deal with the blow-up properties of the positive solutions to the system of heat equations with nonlinear boundary conditions: NEWLINE\[NEWLINE\begin{cases} u_{it}=\Delta_x u_i,\quad i= 1,\dots, k,\quad u_{k+1}:= u_1,\quad x\in\Omega,\quad t\in J,\\ {\partial u_i\over\partial\eta}= u^{p_i}_{i+ 1},\quad x\in\partial\Omega,\quad t\in J,\\ u_i(x,0)= u_{i,0}(x),\quad x\in \Omega,\end{cases}\tag{1}NEWLINE\]NEWLINE where \(J= (0,T)\), \(\Omega\subset \mathbb{R}^N\) is a bounded domain with smooth boundary, \(\eta\) is the unit outward normal vector, \(u_{i,0}(x)\) are nonnegative functions satisfying appropriate compatibility conditions. The main goal of this note is to derive the blow-up rate estimates. The authors give the criteria for the solution to have a finite time blow-up. Moreover, the blow-up set is discussed as well.