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)


1 Answer 1


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}


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