Consider the problem (Nonlinear Schrödinger equation)
\begin{equation}
\left\{
\begin{array}{rl}
 iu\_t + \Delta u\mp u|u|^{\alpha}=0\\
 u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\
\end{array}\right.
\end{equation}
where $N\geq 3$ and $0<\alpha<\frac{4}{N}$.


I consider solution in the strong sense, i.e. $u(t,x)\in C^0([0,T],H^1(\mathbb{R}^N))$ such that


\begin{equation}
u(t)= e^{it\Delta}\varphi \pm i\int\_0^t e^{i(t-s)\Delta}u|u|^{\alpha}(s)ds,
\end{equation}


where $T=T\left(\|\varphi\|\_{H^1}\right)$. And $e^{it\Delta}\varphi(x)$ is the solution of the linear Schrödinger equation:


\begin{equation}
\left\{
\begin{array}{cc}
 iu\_t + \Delta u=0\\
 u(0) =\varphi\in H^{1}(\mathbb{R}^N). \\
\end{array}\right.
\end{equation}


I can prove that such solution exist.


I know that there exist the following dichotomy:


1) $T=+\infty$


2)$T<\infty$ and $\lim\_{t\rightarrow T^{-}} \|u(t)\|\_{H^1}=+\infty.$


I would like to know how to prove this dichotomy.


Maybe it is a more general fact, but in order to not make mistakes I preferred to enunciate the statement in my particular case.


Thank you for your help.


(I asked the same question on <https://math.stackexchange.com/questions/1044796/dichotomy-for-global-existence-or-blow-up-for-solutions-of-evolution-problems> without any answer, I apologize for the non research quality question)



Dichotomy for global existence or blow up for solutions of evolution problems
It is indeed a quite general fact and this is a consequence of the use of the fixed-point theorem.


You have rewritten your equation in the mild form


\begin{equation}
u(t)= e^{it\Delta}\varphi \pm i\int\_0^t e^{i(t-s)\Delta}u|u|^{\alpha}(s)ds,
\end{equation}


which allows you to solve the abstract equation $u = f + F(u)$, $f$ being some function.


The point is : until when can you make use of the fixed-point ? Well, as long as your solution does not blow-up. If it does, the equation does not make sense anymore (at least in the space $C^0([0,T],H^1(\mathbb{R}^N))$ which you consider). If it doesn't, then you can apply again the fixed-point to push a little beyond your previous time of existence.


Notice that such criteria depend on the spaces in which you use the fixed-point : if your space was, say, $L^2([0,T],H^1(\mathbb{R}^N))$ instead, you would get the following blow-up criteria : 


\begin{equation}
\int\_0^T \| u(t) \|\_{H^1}^2 dt = + \infty
\end{equation}


