  This paper estimates the blow-up time for the heat equation $u_t=\Delta u$
with a local nonlinear Neumann boundary condition: The normal derivative
$\partial u/\partial n=u^{q}$ on $\Gamma_{1}$, one piece of the boundary, while
on the rest part of the boundary, $\partial u/\partial n=0$. The motivation of
the study is the partial damage to the insulation on the surface of space
shuttles caused by high speed flying subjects. We prove the solution blows up
in finite time and estimate both upper and lower bounds of the blow-up time in
terms of the area of $\Gamma_1$. In many other work, they need the convexity of
the domain $\Omega$ and only consider the problem with
$\Gamma_1=\partial\Omega$. In this paper, we remove the convexity condition and
only require $\partial\Omega$ to be $C^{2}$. In addition, we deal with the
local nonlinearity, namely $\Gamma_1$ can be just part of $\partial\Omega$.
