finite time blow-up criterion in nonlinear Schrodinger

Question

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?

Answer 1

Let me complete Amir Sagiv's answer by answering parts 1 and 2, and clarifying a bit the comment of Tim Carson.

Let's tackle (2) first: assuming your local wellposedness result, it is impossible to have $u\in C([0,T^*], X)$ but $u\not\in C([0,T^*+\epsilon],X)$ for any $\epsilon > 0$: this is because you can solve the initial value problem starting from $t = T^*$ with $\tilde{u}_0 = u(T^*)\in X$ and, applying the local wellposedness result, arrive at the existence of some $T' > 0$ such that $\tilde{u} \in C([T^*,T^* + T'],X)$ that solves the equation.

Similarly, to answer (1), you see that your local wellposedness result actually implies that if $u(T) \in X$ you can extend to a solution on $C([0,T+\epsilon),X)$.

Morally speaking: that you have a local well-posedess result states that any time in the domain of existence has to be an interior point, and hence that the domain of existence should be open. The main difficulty in most global-existence-type results is in demonstrating that a solution that is known to exist on open intervals of the form $[0,T^*)$ can be extended to intervals of the form $[0,T^*]$.

Answer 2

I can give partial answers to 3-4.

One definition to finite-time blowup is: there's a $T_c>0$ so that $\lim\limits_{t\to T_c ^-} \|u(t,\cdot)\| = \infty$. What is the appropriate norm? When $F(u) = |u|^{2\sigma}u$, for example, there's both an $L^{\infty}$ and $H^1$ blowup for certain values of $\sigma$ and initial conditions in the respective normed space. For more details, see "The nonlinear Schrodinger equation: Self focusing, singular solutions and optical collapse" by Gadi Fibich, specifically chapters 5.5-5.6.

From here it is apparent that there might be local well posedness in the aforementioned norms and a finite time blowup for the same initial conditions. So, it seems to me that the answer to question (4) as I understand it is that you can't generally use local well posedness to prove the global one.

Edit: However, in the defocusing case $F(u) = -|u|^{2\sigma}u$ as well as the focusing subcritical case, $F(u) = |u|^2 u$ and $u:\mathbb{R} \to \mathbb{R}$, it is exactly the same local well posedness with which one proves global existence, see the reference above for details.