Global non-scattering solutions conjecture for the cubic wave equation

Consider the cubic wave equation and its phase-space notation $\boldsymbol{u}(t)=(u(t),\partial_tu(t))$, with $\mathcal{H}^s=\dot H^s(\mathbb{R}^3)\times\dot H^{s-1}(\mathbb{R}^3)$. A solution scatters if it approaches a solution of the linear wave equation as $t\to\infty$. A self-similar solution has the form

$$S(t,x)=\frac{1}{t}\varphi\left(\frac{x}{t}\right).$$

Global non-scattering solutions conjecture. Let $u$ be a solution of the cubic wave equation with the prescribed initial-data problem, defined for $t\in[0,\infty)$ and not scattering to a linear solution as $t\to\infty$. Then there exists a self-similar solution $S$ such that

$$\lim_{t\to\infty}\left\|u(t)-S(t)\right\|_{L^3(\{|x|<t-\sqrt{t}\})}+\left\|\partial_tu(t)+\partial_tS(t)\right\|_{L^{3/2}(\{|x|<t-\sqrt{t}\})}=0.$$

Furthermore, there exists a linear-wave solution $v_L$ with initial data satisfying $\boldsymbol{v}_L(0)\in\bigcap_{1/2<s\leq1}\mathcal{H}^s$ and $\boldsymbol{v}_L(0)\notin\mathcal{H}^{1/2}$, such that the stated asymptotic-linear relation holds. This conjecture proposes a general asymptotic description of global non-scattering solutions; the source does not establish it or determine which self-similar profile must occur.