I am given a non-negative solution $ u\geq 0$ to the heat equation on a bounded open subset $\mathbb{\Omega}$ of $\mathbb{R^n}$ and $t>0,$ so $(x,t) \in \Omega \times (0,\infty).$ I have homogenous Dirichlet boundary conditions, $u(t,x) = 0, x \in \partial \Omega, t>0.$
What can I say about mass conservation? I suspect that $ \frac{d}{dt}\int_{\Omega} u(x,t) dx \leq 0,$ because if I keep the boundary $0$, I expect the total heat of a rod to decline over time. Is this intuition correct? How can I prove it mathematically?
$ u_t(x) > 0$ and $u_t(x \in \partial \Omega )=0$ plus smoothness implies negative outward derivative at the boundary $\partial \Omega$.
With Hodge-Laplacian $\Delta = d*d$, the conversion of the volume integral to the surface integral (by the local definition of d*=div by surface integrals)
$$\partial_t \int_\Omega \ d\Omega u_t(x) = \int_\Omega \ d\Omega\ \partial_t u_t(x) = \int_\Omega \ d\Omega\ \ d* d u_t(x) = \int_{\partial \Omega} \ *d u_t(x) < 0 $$