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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.

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