# Weak derivative under the integral sign

Let $$\Omega$$ be a bounded and regular open subset $$\Omega$$ of $$\mathbb{R}^N$$ and $$u:[0,\infty)\times \Omega\to \mathbb{R}$$ be a smooth function (for example a smooth solution to a PDE). Thus the function $$w=\min(0,u)$$ has weak time derivative given by $$w_t=u_t.1_{\{u \leq 0 \}}$$. Does the following "weak differentiation under the integral sign" holds? $$\frac{d}{dt}\int_\Omega w(t,x)dx=\int_\Omega \frac{\partial}{\partial t}w(t,x)dx,$$ for almost all $$t\geq 0$$.

We know that differentiation under the integral sign holds for $$u$$ because it is smooth. But I am wondering if it also holds for a function like $$w=\min(0,u)$$ which only has a weak derivative.

My guess is that since $$w$$ has a weak derivative in time (real line), then it must be differentiable for almost all $$t\geq 0$$ (which is not necessarily true in higher dimension, for example differentiation with respect to space $$x$$). So $$\int_\Omega w(t,x)dx$$ must also be differentiable for almost all $$t\geq 0$$ and so we can use some kind of dominated convergence theorem.