Improperly posed problems that do have solutions.

  You might think that as long as we specify an infinitely smooth heat flux, this problem would not arise. And indeed, for some (not all) infinitely smooth heat fluxes, there is in theory a solution to the initial value problem for the Laplace equation. The difficulty is that if we make even the slightest, imperceptible kink in such a heat flux distribution, it is singular and the solution suddenly no longer exists. As a result, even the smallest change in the given conditions can completely alter the solution. A unique solution would still require infinite precision in the values of the heat flux.

A problem in which small disturbances can have arbitrarily large effects is called an improperly posed problem. Both the initial value problem for the Laplace equation and the boundary value problem for the wave equation are improperly posed. So is the problem of trying to solve the heat equation backward in time, for similar reasons: since the heat equation smooths singularities, no earlier time solution can give rise to a singularity in the bar at the later starting time. On the other hand, it is perfectly well possible to solve the wave equation backward in time.