Let $u:\Omega \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator.

How can I compute the following distributional derivative?

$$ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx.$$