Take the 2-minute tour ×   consider a parabolic boundary value
problem, for instance  $-\partial_tu+\Delta u=0$, in $\Omega$,
$\partial_\nu u=0$ on $\partial\Omega$,  in a domain
$Q=(0,T)\times\Omega$, where $\Omega\subset R^n$ is bounded. Now
suppose $u$ is smooth and attains its maximum in $(T,x_0)$ for
some $x_0\in\partial\Omega$. Is it possible to say anything
about the normal derivative $\partial_\nu u(T,x_0)$ similarly as
in Hopf's boundary point lemma?  (Hopf's boundary point lemma is
not applicable, since there is no circle tangent to $\partial Q$
at $(T,x_0)$ entirely contained in $Q$!)  I came across this
problem by the following thought:  By classical maximum
principles the maximum can't be attained in $\Omega$, so it has
to be located on the boundary. What if the maximum is at the (in
time) top end? Although the initial distribution is strictly
negative, it could be positive without violating the boundary
condition $\partial_\nu u=0$ on $\partial\Omega$, so there would
be a change in signs. Of co