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Let $V \subset H \subset V^*$ be a Gelfand triple of Hilbert spaces. Take $f,\psi \in L^2(0,T;H)$ and consider the VI: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that

$$u(t) \leq \psi(t) : \int_0^T \langle u' + Au - f, v-u \rangle \geq 0\quad \forall v \in L^2(0,T;V) : v(t) \leq \psi(t).$$

$A$ is some smooth elliptic operator.

Can I expect $u' \in L^2(0,T;H)$, at least with additional assumptions? Could anyone refer me to a reference for this? The only literature is see has the obstacle $\psi$ independent of time, which is not the case here.

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