Is there a maximum principle for the heat equation
$\partial_t u(x,t)=k \partial_{xx}^2 u(x,t)$
for $(x,t)\in[O,L] \times [0, \infty]$?
If $u$ has a maximum it would occur at $t=0$, $x=0$ or $x=L$, just like in the bounded case, but since we can't assume $u$ attains its maximum, I don't know whether it has a maximum principle or how to prove it.