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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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