I have been learning PDEs (more specifically, nonlinear dispersive equations (Schrödinger/wave/ Klein-Gordan equations etc...)) through the harmonic analysis methods. And I have read a couple of papers which handle local well-posedness results (mainly with contraction mapping principle) and global well-posedness results (mainly with conservation laws/persistence of regularity etc...).

Let $X$ be a Banach space of functions (distributions) on $\mathbb R.$ For instance, we consider the nonlinear Schrödinger equation (NLS): $$iu_{t}+\Delta u = F(u), \ u(t_{0}, x)=u_{0}(x)\in X,$$ where $u:\mathbb R \times \mathbb R \to \mathbb C, u_{0}:\mathbb R\to \mathbb C, F:\mathbb C \to \mathbb C$ are functions, and with initial time $t_{0}\in \mathbb R.$

We suppose that the local well-posedness results and blowup criteria hold in $X,$ that is, for the given initial data $u_{0}\in X$ there is a $T^*= T^*(\|u_{0}\|_{X})>t_{0}$ so that the above NLS has unique solution $u\in C([t_{0}, T^*], X);$ and if $T*<\infty,$ then $\limsup_{t\to T^*} \|u(t, \cdot)\|_{X}= \infty.$

**My Questions**:

(I) Can you give an example of some Banach space $X$ where local well-posedness and blowup criteria hold (in the above sense) but one can not extend to a solution for all time (I mean one cannot get global solution)?

(II) Is there any well-known method, which can show there do not exist global well-posedness results?

[Of course the question highly depends on $X$ but I am just looking for examples where we have local well-posedness but do not have global well-posedness, and keen to know of a method (if it exists) with which one can show global result does not exist. I hope this question makes sense and some proper references will be o.k for me.]

Thanks,