Maximal existence time in a solution class

A solution $u$ of a Cauchy problem on $[t_0,T_*)$ , with $T_*\in(t_0,\infty]$ , is maximal in a specified solution class if no solution on a longer interval agrees with $u$ on this interval and belongs to that class. The number $T_*$ is its maximal existence time, and $T_*-t_0$ is its lifespan.

What is fixed

An extension solves the same equation with the same data and with coefficients and forcing that are defined on the longer interval. Maximality is relative to the spatial domain, boundary conditions, and regularity or integrability class.

Norm criteria require a theorem

A finite maximal time does not by definition identify which norm becomes unbounded. A continuation theorem is needed to deduce norm growth from failure to extend. In a problem with local uniqueness, compatible local extensions combine into one maximal solution.