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

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    $\begingroup$ This is too many boundary conditions when $\eta=0$. So as it stands, the problem is not properly formulated. $\endgroup$
    – Michael Renardy
    Commented Aug 14 at 3:14
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    $\begingroup$ Really I would only want that $u(x,T)=f(x)$ for some smooth function f on $\mathbb{R}^d$, not sure what the boundary condition for the corresponding elliptic problem should be tbh... $\endgroup$
    – ABIM
    Commented Aug 14 at 3:25
  • $\begingroup$ I don't understand your notation. If $f$ is only defined on the boundary of the unit cube, how can you have $u=f$ on $\mathbb{R}^d$? In any case, the first approach to such a problem would be to take the Fourier transform with respect to $x$, possibly even starting in 1d for simplicity. This gives you a bunch of independent ODEs which can be explicitly solved and will tell you a lot about the behaviour and about what initial and boundary conditions could work. $\endgroup$
    – mlk
    Commented Aug 14 at 8:40
  • $\begingroup$ @mlk I fixed this, but I still have not managed to find pertrubation bounds. $\endgroup$
    – ABIM
    Commented Oct 17 at 19:02

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