The classic example of an ill-posed parabolic PDE problem is the "backward-in-time heat equation".

Here, if we think of as the temperature in a one
dimensional heat conduction rod, the condition
can be thought of as giving the temperature distribution at
some specific time **t = T**. The PDE problem calls for using
this information, together with the heat balance equation
and the boundary conditions to predict the temperature
distribution at some earlier time, sat **t = 0**. It can be
shown (see Schaum's Outline of PDE, solved problem 4.9) that
if is not infinitely continuously differentiable, then
no solution to the problem exists. If is infinitely
continuously differentiable, then it is shown that the
solution on **0 < t < T** does not depend continuously on the
data, namely .