Blow-up problems for the heat equation with a local nonlinear Neumann boundary condition

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 $\partial\Omega$.