Assume that $u \in W^{1,2}(0,T; W_0^{1,2}(\Omega))$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ and $T <+\infty$.
Let $A \subset \Omega \times (0, T)$ be a Lipschitz domain such that $u(\cdot,t_0) \leq 0$ on $\partial (A \cap \{t=t_0\})$ for a.a. $t_0 \in (0, T)$ in the sense of traces of functions from $W^{1,2}(\Omega)$.
Is it true that $\max(u, 0)|_{A} \in W^{1,2}(0,T;W_0^{1,2}(\Omega))$?
(Here under $\max(u, 0)|_{A}$ I mean a function $w$ such that $w = \max(u,0)$ on $A$ and $w=0$ on $(\Omega \times (0, T)) \setminus A$).
P.S. Analogous property is, of course, valid for functions from $W_0^{1,2}(\Omega)$.
[Added] The question is also actual for the space $W^{1,2}(0,T;L^2(\Omega)) \cap L^2(0,T;W_0^{1,2}(\Omega))$.