This thesis studies the blow-up problem for the heat equation $u_t=\Delta u$ in a $C^{2}$ bounded open subset $\Omega$ of $\m{R}^{n}(n\geq 2)$ with positive initial data $u_{0}$ and a local nonlinear Neumann boundary condition: $\frac{\p u}{\p n}=u^{q}$ on partial boundary $\Gamma_1\subseteq\p\O$ for some $q>1$ and $\frac{\p u}{\p n}=0$ on the rest of the boundary. The motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects. First, we establish the local existence and uniqueness of the classical solution for such a problem. Secondly, we show the finite-time blowup of the solution and estimate both upper and lower bounds of the blow-up time $T^{*}$. In addition, the asymptotic behaviour of $T^{*}$ on $q$, $M_{0}$ (the maximum of the initial data) and $|\Gamma_{1}|$ (the surface area of $\Gamma_{1}$) are studied. \begin{itemize}\item As $q\searrow 1$, the order of $T^{*}$ is exactly $(q-1)^{-1}$.\item As $M_{0}\searrow 0$, the order of $T^{*}$ is at least $\ln(M_{0}^{-1})$; if the region near $\Gamma_{1}$ is convex, then the order of $T^{*}$ is at least $M_{0}^{-(q-1)}/\ln(M_{0}^{-1})$; if $\O$ is convex, then the order of $T^{*}$ is at least $M_{0}^{-(q-1)}$. On the other hand, if the initial data $u_{0}$ does not oscillate too much, then the order of $T^{*}$ is at most $M_{0}^{-(q-1)}$. \item As $|\Gamma_{1}|\searrow 0$, the order of $T^{*}$ is at least $\ln(|\Gamma_{1}|^{-1})$ and at most $|\Gamma_{1}|^{-1}$.If the region near $\Gamma_{1}$ is convex, then the order of $T^{*}$ is at least $|\Gamma_{1}|^{-\frac{1}{n-1}}\Big/\ln\big(|\Gamma_{1}|^{-1}\big)$ for $n\geq 3$ and $|\Gamma_{1}|^{-1}\big/\big[\ln\big(|\Gamma_{1}|^{-1}\big)\big]^{2}$ for $n=2$. If $\O$ is convex, then the order of $T^{*}$ is at least $|\Gamma_{1}|^{-\frac{1}{n-1}}$ for $n\geq 3$ and $|\Gamma_{1}|^{-1}\big/\ln\big(|\Gamma_{1}|^{-1}\big)$ for $n=2$.\end{itemize} Finally, we provide two strategies from engineering point of view (which means by changing the setup of the original problem) to prevent the finite-time blowup. Moreover, if the region near $\Gamma_{1}$ is convex, then one of the strategies is applied to bound the solution from above by $M_{1}$ for any $M_{1}>M_{0}$. For the space shuttle mentioned in the motivation of this thesis, $\Gamma_{1}$ is on its left wing of the shuttle, so the region near $\Gamma_{1}$ is indeed convex. In addition, the relation between $T^{*}$ and small surface area $|\Gamma_{1}|$ is of particular interest for this problem. As an application of the above estimates to this problem, let $n=3$ and $|\Gamma_{1}|\searrow 0$, then the order of $T^{*}$ is between $|\Gamma_{1}|^{-\frac{1}{2}}\Big/\ln\big(|\Gamma_{1}|^{-1}\big)$ and $|\Gamma_{1}|^{-1}$. On the other hand, one of the strategies can be applied to prevent the temperature from being too high.This thesis seems to be the first to systematically study the heat equation with piecewise continuous Neumann boundary conditions. It also seems to be the first to investigate the relation between $T^{*}$ and $|\Gamma_{1}|$, especially when $|\Gamma_{1}|\searrow 0$. The key innovative part of this thesis is Chapter 4. First, the new method developed in Chapter 4 is able to derive a lower bound for $T^{*}$ without the convexity assumption of the domain which was a common requirement in the historical works. Secondly, even for the convex domains, the lower bound estimate obtained by this new method improves the previous results significantly. Thirdly, this method does not involve any differential inequality argument which was an essential technique in the past on the blow-up time estimate.
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