# If $u \le 0$, $u_{xx} - u_t \ge 0$, and $u(2,0) = 0$ on a bounded open set, where must $u$ vanish?

Let $\Omega \subseteq \mathbb{R}^2$ be the open set defined by $\Omega = ((-4,4) \times (-4,4)) \setminus ([-1,1] \times [-1,1])$. For a function $u(x,t) \in C^{2,1}(\Omega)$, let the heat operator $H$ acting on $u$ be given by $Hu = u_{xx} - u_t$.

Suppose now that $u \le 0$ and $Hu \ge 0$ in $\Omega$. Furthermore, assume that $u(2,0) = 0$. Determine the subset of $\Omega$ on which such a function $u$ must vanish.

This is a question from an old PDE qual exam that I am studying. So far, I have determined that $u$ need not vanish on the set $\Omega^+ =\Omega \cap (\mathbb{R} \times (0, \infty))$. This is because we can take $u(x,t) = -G(x,t)$, where $G(x,t)$ is the the fundamental solution to the heat equation:

$G(x,t) = \cases{ (4\pi t)^{-\frac{1}{2}}e^{-\frac{x^2}{4t}} &$t \gt 0$,\cr 0 &$ t \le 0$. }$

Then, in this situation, we have $u \lt 0$ in $\Omega^+$.

I am still trying to figure out how $u$ might behave on $\Omega \cap (\mathbb{R} \times (-\infty, 0]).$ In my PDE course, we did discuss the maximum principle for the heat equation on "cylindrical" domains, which have the form $U \times [0,T)$, for $U$ open in $\mathbb{R}^n$. However, I'm unsure if the principle can be applied to this particular $\Omega$ . Hints or solutions are greatly appreciated!

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The maximum principle would not apply to the entire $\Omega$ at once, but you may get something by using it on subdomains such as $(1,4)\times (-4,4)$. –  user53153 Dec 16 '12 at 16:52