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I need help with this problem. I think that I have to use the maximum principle for the heat equation, but I don't know how.

Let $u$ be a solution of $u_t =u_{xx}$ in the rectangle $S_{T}=(0,1)\times (0,T)$, continuous in the closed set $\bar S_T$. Also suppose that $u_x$ is continuous in $[0,1]\times (0,T]$. Let $0 and $u(x,t)>m$ for $x \in [0,1]$, $t \in (0, t_0]$. Also, let $u(0,t_0)=m$. Prove that $u_x(0,t_0)>0$.

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    @shilov: Take $u(x,t_0)=x^2$. Obviously, \forall\, h>0,\;\frac{u(h,t_0)-u(0,t_0)}{h}=h>0, but why u_x(0,t_0)>0?2014-05-13

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This is known as Giraud-type theorem for parabolic equations. For the details see: abstract at http://link.springer.com/article/10.1007/BF00967266#page-1; full text at http://booksc.org/book/12187461