finite-time blow-up

Finite-time blow-up is when a solution does not just grow large but actually races to infinity at a particular finite value of the independent variable, reaching it in a bounded amount of 'time' rather than taking forever. The solution exists perfectly well up to that instant and then simply ceases to exist as a finite function. It is the dramatic way a maximal interval of existence can end.

It happens when the right-hand side grows faster than linearly in y, so that as the solution gets bigger, it also gets steeper, in a runaway feedback loop. The clean model is y' = y^2, y(0) = 1. Separating variables gives the solution y = 1/(1 - x), which is fine for x < 1 but rockets to infinity as x approaches 1 and is undefined beyond. Compare y' = y, whose solution e^x also grows without bound but stays finite at every finite x — that is mere exponential growth, not blow-up. The difference is that y^2 feeds back on itself hard enough to reach infinity in finite time, while y only manages it in the limit.

The honest and slightly surprising point is that nothing about y' = y^2 looks dangerous: the right-hand side is a smooth polynomial, Lipschitz on any bounded region, so Picard-Lindelof gives a unique local solution with no warning of trouble. Blow-up is a global phenomenon that local theory cannot see. This is the practical reason the maximal interval of existence is a real concept and not a technicality — superlinear growth in a model (in population dynamics, combustion, gravitational collapse) genuinely produces solutions that are valid only up to a finite catastrophe.