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The blow-up rate for the heat equation with a nonlinear boundary condition. (English) Zbl 0735.35014
The authors consider the problem: \(v_ t=\Delta v\) in \(B\times(0,T)\); \(\partial v/\partial n=v^ p\) on \(\partial B\times(0,T)\); \(v(\zeta,0)=v_ 0(\zeta)\), \(\zeta\in\bar B\) where \(B:=\{\zeta\in\mathbb{R}^ n: |\zeta|<1\}\), \(p>1\) and \(v\geq0\) satisfies the boundary condition, is smooth, and has the form \(v_ 0(\zeta)=u_ 0(|\zeta|)\) for some \(u_ 0: [0,1]\to\mathbb{R}\). It is assumed, that the first four derivatives of \(u_ 0\) are non-negative, that \(u_ 0(1)^{p-1}\geq2N\) if \(N>1\) and \(u_ 0(1)>0\) if \(N=1\). It then follows that the required solution \(v\) has the radially symmetric form \(v(\zeta,t)=u(|\zeta|,t)\).
The initial boundary value problem governing \(u\) is given and it is pointed out that, under the assumptions on \(u_ 0\), the solution \(u\) blows up in finite time \(T=T(u_ 0)\). Furthermore it is known that \(u(1,t)\to\infty\) as \(t\to T\). The paper gives an explicit description of the behaviour of \(u\) near \((1,T)\). The results obtained are in contrast with those obtained when the nonlinearity appears in the equation.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
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