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 . 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 . 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 does not depend continuously on the data, namely .
For simplicity, choose .
If is not infinitely continuously differentiable: Write , the initial state (temperature). Then the function
will solve the problem, provided the are such that
But the above series converges uniformly to an infinitely differentiable function of , whatever the . It follows that no solution exists when is not infinitely differentiable.
If is infinitely continuously differentiable: We will show that the solution does not depend continuously on the data. If, for instance,
the unique solution to the problem is
For large , on the one hand, becomes uniformly small; that is, the data function differs by as little as we wish from the data function , to which corresponds the solution . On the other hand, grows with ; i.e., the solution does not remain close to . Thus, there is no continuity of dependence on the data.