Single-particle billiards systems in a domain with corners, or multi-particle billiards in a domain with smooth boundary, can exhibit singularities in finite time. (The former phenomenon is well known; for an example of the latter, see e.g. section 1 of Charles Radin's article "Dynamics of Limit Models", available at http://www.ma.utexas.edu/users/radin/papers/limitmodels.pdf.)
Can single-particle billiards exhibit such catastrophes in a domain with smooth boundary?
(If the answer depends on the degree of smoothness, e.g. if the answer is different for $C^1$ vs $C^\infty$, then interpret my question as "For what forms of smoothness is it the case that ..., and for what forms of smoothness is it not the case that ...?")
Note that this is not the same as asking the corresponding question about the discrete billiards map. The discrete map can be iterated unboundedly many times, but if the return-time to the boundary shrinks quickly, "time infinity" under discrete (return-map) dynamics could correspond to a finite-time catastrophe in the flow.
Note also that I am not asking whether billiards flow is well-defined for a set of initial conditions of full measure; I am asking whether it is well-defined for ALL initial conditions. (So please don't add an ergodic-theory tag to my post!)
Note also that the catastrophe phenomenon is not the same as the ill-definedness of the velocity at the moment of rebound. This ill-definedness does indeed make it slightly tricky to define billiard flow, but this is merely a technical problem that is easily surmountable. Extending the dynamics beyond a catastrophe is more problematic; it entails breaking symmetry and/or introducing discontinuities.
Hopefully I have anticipated all misunderstandings of my question that are likely to arise.