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Stability of perturbation of an elliptic problem into a parabolic PDE

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 …