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6 Well Posed PDE Problems     continued...

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 .