Proof of an identity in a Lebesgue-Bochner space

Question

Let $u \in L^2(0,T;H^1(\Omega))$ such that $\partial_t u \in L^2(0,T;(H^1(\Omega))^*)$ and define the following function $$ f_\epsilon(s)=\begin{cases}s \;\; \text{if} \hspace{0.4cm} \vert s\vert \leq \epsilon,\\ \epsilon \;\; \text{if} \hspace{0.4cm} s > \epsilon,\\ - \epsilon \;\; \text{if} \hspace{0.4cm} s < -\epsilon \end{cases} $$ Prove that $\partial_t f_\epsilon(u) \in L^2(0,T;H^1(\Omega)^*)$ and that the following identity holds $$ \partial_tf_\epsilon(u)=\partial_tu\; \mathbf{1}_{\vert u \vert < \epsilon} $$ I know that the above identity holds true if we replace $\partial_t$ with $\nabla,$ but I'm not able to prove it for $\partial_t.$

Answer 1

In the publication https://dx.doi.org/10.14712/1213-7243.2015.168 it is shown that the time derivative of the positive part $\max(u, 0)$ (defined in a pointwise way) may not belong to $L^2(0,T;H^1(\Omega)^\star)$. Since your function is very similar (truncation at $\pm \epsilon$ instead of truncation at $0$), I believe that $\partial f_\epsilon(u) \in L^2(0,T;H^1(\Omega)^\star)$ can fail.