Classification of minimal-mass blowup solutions

Let $u_0 \in L_x^2(\mathbb{R}^d)$ satisfy $M_0(u_0)=M_0(Q_0)$. Suppose $u$ is a maximal-lifespan solution of the equation in the source with maximal lifespan $I$, and suppose it blows up in the sense that

$$\|u\|_{L_{t,x}^{\frac{2(d+2)}{d}}(I\times\mathbb{R}^d)}=\infty.$$

Here $Q_0$ is the ground-state profile, $M_0$ denotes the mass, and a soliton solution is a global solution of the displayed form below. A pseudo-conformal transformation of the soliton solution is a solution of the second displayed form, with the parameters specified there.

Classification of minimal-mass blowup solutions. Only the following two cases occur:

1. $I=(-\infty,\infty)$ and there exist $\lambda>0$, $\gamma\in[0,2\pi]$, and $(\tilde{x},\xi)\in\mathbb{R}^d\times\mathbb{R}^d$ such that

$$u(t,x)=e^{i\gamma-it|\xi|^2}e^{i\lambda^2t}e^{ix\cdot\xi}\lambda^{\frac d2}Q_0\left(\lambda(x-2t\xi)-\tilde{x}\right).$$

2. $I=(-\infty,T)$ or $(T,\infty)$ for some $T\in\mathbb{R}$, and there exist $\lambda>0$, $\gamma\in[0,2\pi]$, and $(\tilde{x},\xi)\in\mathbb{R}^d\times\mathbb{R}^d$ such that

$$u(t,x)=\frac{\lambda^{\frac d2}}{|T-t|^{\frac d2}}e^{i\gamma}e^{\frac{i|x-\xi|^2}{4(t-T)}}e^{i\frac{\lambda^2}{t-T}}Q_0\left(\frac{\lambda(x-\xi)-(T-t)\tilde{x}}{T-t}\right).$$

The preceding results establish only sequential convergence, whereas this conjecture asserts a complete classification of all minimal-mass blowup solutions and their precise time dependence.