Consider nonlinear Schrodiner equation (NLS):
$$i\partial_t u + \Delta = \pm |u|^{p-1}u, u(x,0)=u_0 \in H^1(\mathbb R^d)$$ where $u:\mathbb R^{d+1} \to \mathbb C, u_0:\mathbb R^d \to \mathbb C, 1<p<\infty.$
We put $$E(u(t))= \frac{1}{2} \int_{\mathbb R^d} |\nabla u (x,t)|^2 dx \pm \frac{1}{p+1} \int_{\mathbb R^d} |u(x,t)|^{p+1} dx.$$
I think this is true $E(u(t))= E(u(0))$ for all $t\in \mathbb R$ see this. And so my understanding is $E$ is a constant function of $t.$ Suppose that $W(x,t)=W(x)$ is any solution of the above NLS and $W\in H^{1}.$
My Question: Is it possible that $E(u_0)< E(W)$? How should I view energy function? Is it constant.
[I am thinking this is not possible but I am sure some where I am wrong due to the following motivation]
Motivation: See the hypothesis of Theorem 1, p.5