= Backward heat equation
{title2=$u_t=-D u_{xx},\quad D>0$}
A <Fourier mode> has multiplier $e^{Dk^2t}$, amplifying arbitrarily small high-frequency data. For a fixed positive time $t_0$, initial modes $e^{-Dn^2t_0/2}\sin(nx)$ 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.
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