The conormal blow-up conjecture for self-similar soliton solutions

Let $\alpha$ be sufficiently close to zero, let $\mathfrak{M}_\alpha$ be a blow-up of the compactification of Minkowski space, and let $W$ denote the soliton profile. A function is conormal on $\mathfrak{M}_\alpha$ if it has the conormal regularity associated with this blown-up compactification. Conormal blow-up conjecture. For $\alpha$ sufficiently small, there exists a blow-up $\mathfrak{M}_\alpha$ of the compactification of Minkowski space such that the main equation admits a solution $\phi$ conormal on $\mathfrak{M}_\alpha$ and satisfying

$$\left\lVert \phi(t,\cdot)-t^{\alpha/2}W(t^{\alpha}\cdot)\right\rVert_{L^\infty}\to0\quad\text{as }t\to\infty, \qquad \partial_u(r\phi)|_{\mathcal{I}}=0.$$

The paper describes this as a natural conjecture concerning the asymptotic behaviour of the non-unique global solutions constructed by Donninger and Krieger; its resolution is not supplied here.