# 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. If possible, I would like to ask for a reference addressing such a result.

• Hint: try integrating both sides from $0$ to $t$ and applying Fubini's theorem. – Nate Eldredge Aug 12 at 22:20
• I think you need to assume a bit more for the classical result (about $u$), otherwise examples such as $u(t,x)=x\sin (t/x)$ on $x\in\Omega=(0,1)$ become problematic. – Christian Remling Aug 12 at 22:26