Chapter 6 - Some Improperly Posed Cauchy Problems

Summary

Certain physical phenomena lend themselves to modeling by non-standard Cauchy problems for abstract differential equations, such as the reversed Cauchy problem for abstract parabolic equations (Section 6.2) or the second order Cauchy problem in Section 6.5, where one of the initial conditions is replaced by a growth restriction at infinity. If these problems are correctly formulated, it is shown that they behave not unlike properly posed Cauchy problems: solutions exist and depend continuously on their data.

IMPROPERLY POSED PROBLEMS

Many physical phenomena are described by models that are not properly posed in any reasonable sense; certain initial conditions may fail to produce a solution in the model (although the phenomenon itself certainly has an outcome) and/or solutions may not depend continuously on their initial data. One such phenomenon is, say, the tossing of a coin on a table. Assuming we could measure with great precision the initial position and velocity of the coin, we could integrate the differential equations describing the motion and predict the outcome (head or tails); however, it is obvious that for certain trajectories, arbitrarily small errors in the measurement of initial position and velocity will produce radical changes in the predicted outcome. In studying phenomena like these we are then forced to abandon the deterministic point of view and to obtain information by other means (e.g., by probabilistic arguments). This was known to the creators of probability theory and was explicitly stated by Poincaré almost a century ago.