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