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The blow-up rate for a system of heat equations with nonlinear boundary conditions. (English) Zbl 0941.35008

The authors establish the blow-up rate and locate the blow-up set of some positive solutions of the system \(u_t=\Delta u\), \(v_t=\Delta v\) for \(x\in B_R=\{y:|y|<R\}\), \(t>0\), complemented by the boundary conditions \(\partial u/\partial\nu=v^p\), \(\partial v/\partial\nu=u^q\). Here \(p,q>0\), \(pq>1\), and the initial conditions \(u_0,v_0\) are assumed to be radially symmetric, positive, to satisfy the compatibility conditions and \(\Delta u_0,\Delta v_0\geq 0\). The blow-up rate of \((\max u(\cdot,t),\max v(\cdot,t))\) is of order \(((T-t)^{-\alpha/2},(T-t)^{-\beta/2})\), where \(T\) is the blow-up time, \(\alpha=(p+1)/(pq-1)\), \(\beta=(q+1)/(pq-1)\) and blow-up occurs only at the boundary. Similar results for \(p,q\geq 1\) were obtained in [K. Deng, Z. Angew. Math. Phys. 47, No. 1, 132-143 (1996; Zbl 0854.35054)].

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)

Citations:

Zbl 0854.35054
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References:

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