Let $\Omega$ be open and sufficiently smooth, $0\leq \tau \leq T$ and consider $u \in L^2(0,T; H^1(\Omega)), \partial_t u \in L^2(0,T;L^2(\Omega))$.
Can we conclude $$\int_0^\tau||u(\tau)-u(t)||^2_{L^2(\Omega)}dt\leq C\tau^2\int_0^\tau||\partial_tu(t)||^2_{L^2(\Omega)}dt \quad ?$$
If one could speak about $s\mapsto u(s,y)$ for fixed $y$, and prove those to be in $H^1(0,T)$, then one could use absolute continuity, but this need not to hold here... Do you know if this is true, or have a counterexample in mind?