Heat equation: Energy $\int_{\Omega}u(t, x) dx$ is decreasing with positive initial data and $u=0$ on the boundary

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:

1. Using the divergence theorem, I can start like $$U'(t) = \int_{\Omega} u_t(t, x) \,\Bbb dx = \int_{\Omega} \Delta u(t, x) \,\Bbb dx = \int_{\partial\Omega} \nabla u(t, x) \cdot \mathbf{n} \,\Bbb dx = \ldots ?$$
2. 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 $$(t,x) \in (\{0\}\times\Omega) \cup ([0,t]\times \partial \Omega)$$.
3. 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.
4. 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.