I am trying to understand the philosophy for finite time blow-up criterion and global existence for ODE/PDEs, any suggestion and comments will be useful to me. I hope this question is OK for MO.

Consider nonlinear Schrodiner equation (NLS):

$$i\partial_t u + \Delta + F(u)=0, u(x,0)=u_0$$ where $u:\mathbb R^{d+1} \to \mathbb C, u_0:\mathbb R^d \to \mathbb C, F: \mathbb C \to \mathbb C$

Suppose $X$ is a nice Banach space ( for instance some Sobolev space $H^s(\mathbb R^d)$ or so).

We also assume the following theorem is true. Theorem (Local wellposedness result): Assume that $u_0\in X.$ Then there exist unique $T= T(\|u_0\|_X)>0$ such that NLS has a uniqe solution in $u\in C([0, T], X).$ [Note. One way of proving this kind of "Local well posedness result" is to use Banach contrcrtion pricniple]

Now my Questions are:

(1) What is a general method to show that the solution $u$ also in $C([0, T+\epsilon], X)$ for some $\epsilon >0$? How long we one can continue this process? (I mean, Can we also show $u\in C([0, T+\epsilon +r], X)$ for some $r>0$?)

(2)When one can determine that $u\in C([0, T^*], X)$ but $u\notin C([0, T^*+\epsilon], X)$ for any $\epsilon >0$? (Example?)

(3)What is a finite-time blow up? (My Vague understanding is that: if we can show $u\in C([0, T^*], X)$ but $u\notin C([0, T^*+\epsilon], X)$ (for any $\epsilon >0$) then the solution $u$ has finite time blow-up. I may be wrong, kindly correct me.)

(4) If we have $u\in C([0, T], X)$ (local wellposedness), then what is the method to show that $u\in C(\mathbb R, X)$ (global well posedness)? Is there any role of finite time blow-up to in this kind of method?