Is the following a correct proof? It is easier than a proof I have been provided with, but I feel like it is wrong.
Prop: If $ u$ satisfies $ u_{t} = \sum_{i=1}^{n} u_{x_{i} x_{i} } $ on $\ D \times [0,T] $ where $ D $ is some open domain , then u attains its maximum on $\ \partial D$
Proof: By Heine-Borel, $ u $ attains its maximum on $\partial D \cup D $ .
Suppose $u$ attains its maximum at $ x_0 \in D $. At $x_0$ then $u_t = 0 $ (by Fermat), and $ u_{x_{i} x_{i}} < 0$ (as the Hessian is negative definite).
This contradicts our heat equation and therefore the maximum must be attained in $ \partial D $.
I think the mistake is where I claim $u_{x_{i} x_{i}} < 0$, so maybe I am confused about the Hessian.
Best Answer
Corrections:
$u$ attains maximum in $K=(D\cup\partial D)\times [0,T]$, if $D$ is bounded.
$u$ does not attain maximum at $x_0$ but at a point of the form $(x_0,t_0)$.
It $(x_0,t_0)$ is an interior point, i.e., $(x_0,t_0)\in D\times (0,T)$, then $u_t=0$ and $u_{x_ix_i}\le 0$.
So, you do not have sufficient amount of evidence to show that $u$ can attain maximum on the boundary.
Hint. You need to consider $u_\varepsilon(x,t)=u(x,t)+\varepsilon|x|^2$.