I have seen many countre examples concerning the instability and the ill-posedness of the backward heat equation, but all these examples are done in the $||.||_{\infty}$. My questions are:
1) Is the backward heat equation is well-posed in the $||.||_2$ norm (in the Sobolev space setting)?
2) Is it stable (in the Sobolev space setting)?
No. Denote $Z(x,t)=(4\pi t)^{-1/2}E^{-x^2/(4t)}$ the fundamental solution of the heat equation. Function $\psi(x)=Z(x,1)=(4\pi )^{-1/2}E^{-x^2/4}$ belongs to $L_2(\mathbb R^2)$. Considering $Z$ as a solution of the Cauchy problem for the backward heat equation we have $$ \int_{-\infty }^{\infty } Z^2(x,t) \, dx=\frac{1}{\sqrt{8 \pi t}} $$ which tends to infinity as $t\to0+$.