Give a direct proof that if U is bounded and $ u \in C_{1}^2(U_{T}) \cap C(\overline U_{T})$ solves heat equation, then $\max_{\overline U_{T}} u$= $\max_{\tau_{T}}u$
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Have you tried considering the contrary? It may be helpful to first prove that if $u_t -a u_{xx}\leq 0$, and all boundary conditions are $\leq 0$, then $u(x,t)\leq 0$. Then, do the same for greater than or equal to zero, then you can find a maximum principle by contradiction. – Jeremy Upsal Nov 22 '13 at 02:17
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Actually there is a hint for this question: Define $u_{\epsilon}=u-\epsilon t$ for $\epsilon \gt 0$ and show $u_{\epsilon}$ can not attain its maximum over $\overline U_{T}$ at a point in $U_{T}$@JeremyUpsal – Yang Nov 22 '13 at 02:30