local versus global solution

This is the distinction between a solution that is only guaranteed to live on a small interval around the start (local) and one that lives on the whole interval you actually care about — often all real numbers, or all positive time (global). The existence theorems are inherently local: they hand you a solution near x0 and nothing more. Whether that solution extends to a global one is a separate question with a separate answer, and it is easy to assume more than the theorem gives.

Make the difference concrete. A local solution exists on some (x0 - h, x0 + h) and is exactly what Peano and Picard-Lindelof promise. A global solution exists on the entire domain of interest. The two coincide for many friendly equations but part ways dramatically for others: y' = y^2, y(0) = 1 has a perfectly good local solution, yet globally it only reaches x = 1 before blowing up, so it is local-only. The bridge between the two is continuation — extend the local solution as far as it goes — and the obstruction to going global is exactly blow-up or running off the edge of the domain. A standard sufficient condition for global existence is that f grows at most linearly in y (so feedback cannot run away); then no blow-up occurs and the local solution extends for all time.

The practical upshot is a discipline of caution. When a theorem or a textbook says 'a solution exists,' it almost always means locally; reading it as 'for all time' is a common and consequential error, since a model can be perfectly well-posed locally and still describe a process that ceases to exist (a population that explodes, a structure that fails) at a finite moment. Knowing whether your solution is local or global tells you whether the model's predictions are trustworthy over the horizon you plan to use them.