Fix some $f\in H^1(\partial (0,1)^d)$.  
Let $\eta\ge 0$ and for each such $\eta$ consider the solution $u^{\eta}$ to the solution PDE
$$
\begin{cases}
\Delta u & = u_t - \eta u_{tt} \mbox{ on } \mathbb{R}^d\times[0,T] \\
u(x,T)& =f(x) \mbox{ on }\mathbb{R}^d.
\end{cases}
$$

Are there any stability bounds for
$$
\|u^{\eta}-u^0\|_{L^2(0,1)^d}\le \mbox{ some function of }\eta \mbox{ and of }  \|f\|_{H^2(\partial (0,1)^d)}?
$$