Short of having actual experimental evidence, it is hard to say what physical laws may differ in the presence of non-globally hyperbolic spacetimes, so for the most part, in the analysis of non-globally hyperbolic spacetimes, it is assumed that the same laws apply. Of course, this doesn't mean that the behaviours are similar to globally hyperbolic spacetimes, as there are a lot of assumptions we usually make that do not hold for general spacetimes.
Not much changes for the basics of classical physics : you solve the equations of motion given by the Lagrangian given some initial conditions. Things get a bit tricky there, though : for a start, defining boundary conditions is not trivial as there may be no spacelike hypersurface. Given some initial conditions, it may not be possible to get a unique solution. There may be several solutions satisfying it or even none (which isn't a good sign). If the spacetime lacks time orientability, you may not be able to define separate retarded and advanced waves which is generally important for EM.
Some non-globally hyperbolic spacetimes do allow hypercomputations (assuming fairly unphysical things of course, like a machine that can run for an arbitrarily long time and infinite memory), and more generally hypertasks. They are called Malament-Hogarth spacetimes. A Malament-Hogarth spacetime is one such that there exists an event $p$ such that a curve $\gamma$ of infinite proper time exists entirely in the past of $p$, that is,
$$\exists p, \exists \gamma, \gamma \subset I^-(p), l_\gamma = \infty$$
It can be shown that every M-H spacetime is non-globally hyperbolic. On the other hand, it's not a terribly useful construct as information can't actually be communicated to $p$ : any signal from $\gamma$ to $p$ will be infinitely blueshifted, making it unlikely to be a well-defined example of a spacetime with the backreaction.