Minimal travelling-wave selection conjecture for the free-boundary PDE

Let $(u,L)$ solve the free-boundary problem, and define

$$U(x,t):=\int_x^{\infty}u(y,t)\,dy,\qquad U_0(x):=\int_x^{\infty}u_0(dy).$$

Let $\Pi_{\min}(x):=\int_x^{\infty}\pi_{\min}(y)\,dy$ denote the integrated minimal travelling wave. Minimal travelling-wave selection conjecture. The following are equivalent:

1. $\displaystyle \limsup_{x\rightarrow\infty}\frac{1}{x}\ln U_0(x)\leq -\sqrt{2}$;
2. $\displaystyle \limsup_{t\rightarrow\infty}\frac{L_t}{t}\leq \sqrt{2}$;
3. $\displaystyle \lim_{t\rightarrow\infty}\frac{L_t}{t}=\sqrt{2}$;
4. $U(L_t+x,t)\rightarrow\Pi_{\min}(x)$ uniformly in $x$ as $t\rightarrow\infty$.

This conjecture characterises the sufficiently light-tailed initial conditions that select the minimal travelling wave. The equivalence of the tail condition, minimal asymptotic speed and uniform convergence of the profile remains open.