Let $u(t,x)$ be solution to some nonlinear PDE. (like Schrödinger, wave etc.)
We know that there are many well-posedness problems. Especially for the local well-posedness theory, some of them showed maximality of solution, which existing time is no longer extended to bigger time, and solution is blow-up, which solution norm of solution space is blowing up.
When proving blow-up, the thing 'we can find maximal solution $u$' is good fact since we may show it by contradiction by assuming that norm of solution is finite. However, what if the existence of maximal time is not guaranteed?
Here are may question.
$1.$ Proving local well-posedness implies that local time $T$, which depends on initial datum, is maximal?
$1$-$1.$ Or, does it imply existence of maximal time $T^*$?
$2.$ The following blow-up criterion,
Assume that there is a unique solution to some PDE, $u$ such that $u\in C([0,T], X)$ for some given initial data $u_0 \in X$. If $T<\infty$, then $\lim_{t\rightarrow T}\|u(t)\|_{X}=\infty$ (or $\lim\sup_{t\rightarrow T} \|u(t)\|_{X}=\infty$).
can be proved without assumption $T$ is maximal?
