From David Borthwick - Introduction to Partial Differential Equations, Exercise 6.3b (paraphrased),
Let $\Omega \subset \Bbb R^n$ be a bounded domain with piecewise $C^1$ boundary. Suppose $u(t,x)$ satisfies the heat equation $$u_t - \Delta u = 0, \\ u(0,x) > 0 \quad\text{for } x\in \Omega, \\ u(t,x) = 0 \quad\text{for } x\in \partial\Omega.$$ Define the total thermal energy at time $t$ by $$U(t) = \int_{\Omega}u(t,x)\,\Bbb dx.$$ Show that $U(t)$ is decreasing.
My attempts so far:
- Using the divergence theorem, I can start like $$U'(t) = \int_{\Omega} u_t(t, x) \,\Bbb dx = \int_{\Omega} \Delta u(t, x) \,\Bbb dx = \int_{\partial\Omega} \nabla u(t, x) \cdot \mathbf{n} \,\Bbb dx = \ldots ?$$
- By the maximum principle, it should end like $$\ldots = \int_{?} -u(t,x) \,(?)\,\Bbb dx \leq \int_{?} -\min_{[0,t]\times \bar{\Omega}} u(t,x) \,(?)\,\Bbb dx \leq 0$$ since $u(t,x)\geq 0$ for $(t,x) \in (\{0\}\times\Omega) \cup ([0,t]\times \partial \Omega)$.
- For the missing steps in the middle, I get stuck because I can no longer “integrate by parts” to bring $\nabla u$ “up” to $u$. Also, the negative sign seems to be crucial to prove $U$ is decreasing, but I have no idea where it could come from.
- Alternatively observe that $u_t$ also satisfies a heat equation. Maybe we can use the maximum principle on $u_t$ directly, but we don’t know $u_t(0,x)$ for $x\in \Omega$.
Are there more things to observe or are these the right tracks? Preferably only elementary results related to heat equation would be used. Any help is appreciated.
Have you tried the Hopf's lemma? (link: https://en.wikipedia.org/wiki/Hopf_lemma)