Set $u_0\in H^1 (\mathbb{R} ^N)$ and $1<\alpha < \frac{N+2}{N-2}$. I want to show that there exists $\varepsilon > 0$ s.t. if $\Vert u _0 \Vert _ {H^1} < \varepsilon$, then there is global solution (defined for all $t\in\mathbb{R})$ of $$ u_t = i (\Delta u + \vert u \vert ^{\alpha -1} u) $$ $$ u(\cdot,0)=u_0 $$

I have proved the existence of solution (local). Now I want to prove that is in fact global. In order to do that, I thought to use the blow-up alternative, that is, I just have to show that $\Vert u \Vert _{H^1(\mathbb{R}^N)}$ is bounded, where $u$ is a $H^1-$solution. It is clear that $ \Vert u \Vert _{H^1(\mathbb{R}^N)} = \Vert u \Vert _{L^2(\mathbb{R}^N)} + \Vert \nabla u \Vert _{L^2\mathbb{R}^N)}$.

By the mass conservation, $\Vert u \Vert _{L^2(\mathbb{R}^N)} = \Vert u _0\Vert _{L^2(\mathbb{R}^N)} $. By the energy conservation, $ \frac{1}{2}\Vert \nabla u\Vert_{L^2} ^2 - \frac{1}{\alpha + 1} \Vert u \Vert_{L^{\alpha+1}} ^{\alpha +1} \equiv E\in\mathbb{R} $ therefore, by Sobolev embedding, $$ \Vert \nabla u\Vert_{L^2} ^2\leq 2E + \frac{2}{\alpha + 1} \Vert u \Vert_{L^{\alpha+1}} ^{\alpha +1}\leq 2E + \frac{2C}{\alpha + 1} \Vert u \Vert_{H^1} ^{\alpha +1}$$

If I show that $\Vert \nabla u\Vert_{L^2} ^2 \leq 2E + \frac{2C}{\alpha + 1} \Vert u \Vert_{H^1} ^{\alpha +1} \leq \text{constant}$ I finish.
I have the assumption that **$\Vert u _ 0 \Vert _{H^1}$ is small, does it implies that $\Vert u \Vert _{H^1}$ is also small?**
As I am reading on Internet, I found something called Lyapunov stability.
Maybe is useful here, can anyone explain how could be used in this situation? Thanks in advance. Any idea is welcome!