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 $u_x(0,t_0)>0$.