In Evan's proof for the Weak maximum principle
He simply assumes that $u(x_0, t_0) > 0$.
Why can he make this assumption? I have spent some time on this but unfortunately I don't see it. Does it follow from some result for parabolic equations or am I just missing something?
By continuity, $u$ assumes its max on $\bar U_T$. If it is assumed on the parabolic boundary, there is nothing to prove. If it is assumed inside and it is $\le 0$ there is nothing to prove either because the max on the right hand side is $\ge 0$. The only case to be considered is that of a positive max inside, and that actually never happens, as shown.