For nonlinear Schrodinger equation$$\begin{cases}iu_t+\Delta u+|u|^\alpha u=0\\u(0)=\phi\in H^1(\mathbb R^d)\end{cases}$$ where $\alpha>\frac 4d$.
In Christ, Colliander, Tao's paper Ill-posedness for nonlinear Schrodinger and wave equations, page 3, they mentioned that by using scaling $u(t,x)\mapsto\lambda^{-\frac2\alpha}u(\lambda^{-2}t,\lambda^{-1}x)$ and the fact that finite time blow-up solution exists, one can construct an initial data in $L^2(\mathbb R^d)$ such that it doesn't have local well-posedness in arbitrary small time interval.
My question is, how does such construction be done? Can we directly take the sum $\phi=\sum_{j=1}^\infty \phi_j$ where $\phi_j(t,x)=\lambda_j^{-\frac2\alpha}\phi_0(\lambda_j^{-2}t,\lambda_j^{-1}x)$ with $\lambda_j\to\infty$ fast enough? Since there is nonlinearity interference the solution.
