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 $$ \|u^{\eta}-u^0\|_{L^2(0,1)^d}\le \mbox{ some function of }\eta \mbox{ and of } \|f\|_{H^2(\mathbb{R}^d)}? $$