Let $\eta\ge 0$ and, for each such $\eta$ consider the solution operator $T_{\eta}:H^1(\partial (0,1)^d)\to H^1((0,1)^d)$, sending any $f\in H^1(\partial (0,1)^d)$  to the solution PDE
$$
\begin{cases}
\Delta u & = u_t - \eta u_{tt} \mbox{ on } (0,1)^d \\
u& =f \mbox{ on } \partial (0,1)^d.
\end{cases}
$$

So when $\eta=0$ we have a solution operator to a parabolic PDE, but even for small positive values of $\eta$ we are looking at a solution operator to an Elliptic PDE.


Are there estimates on:
$$
\|T_{\eta}(f)-T_0(t)\|_{\infty} \le \mbox{ some function of }\varepsilon \mbox{ and } \|f\|_{H^1(\partial (0,1)^d)}?
$$