Backward heat equation ($u_t=-D{u}_{xx},D>0$)

A Fourier mode has multiplier $e^{D{k}^{2}t}$, amplifying arbitrarily small high-frequency data. For a fixed positive time $t_0$, initial modes $e^{-D{n}^2{t}_0/2}\sin (nx)$ tend to zero in $L^2$ while their values at $t_0$ grow without bound. This violates continuous dependence on initial data and gives an ill-posed problem in ordinary unweighted spaces. An analytic-data restriction or physical short-scale regularization changes that conclusion.