I am trying to understand the proof of Theorem 3.9, p.9.
We consider NLS $$i\partial_t u + \Delta + |u|^{4/(d-2)}u=0, u(x,0)=u_0 \in H^1(\mathbb R^d)$$ where $u:\mathbb R^{d+1} \to \mathbb C, u_0:\mathbb R^d \to \mathbb C$
Assume that $\int_{\mathbb R^d} |\nabla u_0(x)|^2 dx \leq (1-\delta) \|\nabla W \|^2_{L^2}$ and $\int_{\mathbb R^d} |\nabla u_0(x)|^2 - |u_0(x)|^{2d/d-2} \geq \delta \int_{\mathbb R^d} |\nabla u_0(x)|^2$ for some fix $\delta >0$.
Now, let $I$ be the maximal interval of existence of the above NLS.
My Question: How to show $\int_{\mathbb R^d} |\nabla u(x, t)|^2 dx \leq (1-\delta) \|\nabla W \|^2_{L^2}$ and $\int_{\mathbb R^d} |\nabla u(x, t)|^2 - |u(x, t)|^{2d/d-2} dx \geq \delta \int_{\mathbb R^d} |\nabla u(x, t)|^2 dx$ for all $t\in I.$ In other words, if these inequality isinequalities are true at time $t=0$, then why itthey are true for all time time $t\in I$?
(Why this is call energy trapping? Because $E(u(t))\geq 0$?)
[My confusion: The author gives the hint that use continuity argument, what is the meaning of continuity arguement and how to apply here.]
