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The blow-up rate for a system of heat equations with non-trivial coupling at the boundary. (English) Zbl 0870.35050

Let \(P=(p_{ij})\) be a matrix of nonnegative entries such that \(p_{11}<1\), \(p_{22}<1\) and \(\text{det}(P-Id)<0\). Let \((\alpha_1,\alpha_2)\) be a vector of negative entries such that \((P-Id)\) \((\alpha_1,\alpha_2)^T= (-1,-1)^T\). The paper establishes that any positive solution of the problem \(\partial_tv_i= \partial^2_rv_i+ (N-1)r^{-1}\partial_rv_i\) with \(\partial_rv_i(0,t)= 0\), \(\partial_rv_i(1,t)= v^{p_{i1}}_1(1,t)v^{p_{i2}}_2(1,t)\) and \(v_i(r,0)= v_{i0}(r)\) for \(i\in\{1,2\}\), \(r\in[0,1]\), \(t\in[0,T[\), and \(v_{i0}\in C^3\), such that \(\partial_rv_i\), \(\partial_tv_i\), \(\partial_t\partial_rv_i\geq 0\), satisfies \(c\leq v_i(1,t)(T-t)^{-\alpha_i/2}\leq C\) as \(t\to T\).

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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