## Blow-up analysis for a system of heat equations coupled through a nonlinear boundary condition.(English)Zbl 0980.35075

The authors deal with the blow-up properties of the positive solutions to the system of heat equations with nonlinear boundary conditions: $\begin{cases} u_{it}=\Delta_x u_i,\quad i= 1,\dots, k,\quad u_{k+1}:= u_1,\quad x\in\Omega,\quad t\in J,\\ {\partial u_i\over\partial\eta}= u^{p_i}_{i+ 1},\quad x\in\partial\Omega,\quad t\in J,\\ u_i(x,0)= u_{i,0}(x),\quad x\in \Omega,\end{cases}\tag{1}$ 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.

### MSC:

 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs

### Keywords:

blow-up rate estimates; positive solutions; blow-up set
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### References:

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