Formal asymptotic conjecture for front propagation with evolving dispersion

Let $f(t,x,\theta)$ be the density solving the rescaled sexual-population equation

$$\partial_t f=B[f]-\varrho f+\theta\Delta_x f,$$

with trait $\theta\geq 1$, population size $\varrho(t,x)=\int_1^\infty f(t,x,\theta)\,\mathrm{d}\theta$, reproduction operator

$$B[f](t,x,\theta)=\iint_{(1,\infty)^2}\mathcal G_\lambda\left[\theta-\frac{\theta_1+\theta_2}{2}\right]f(t,x,\theta_1)\frac{f(t,x,\theta_2)}{\varrho(t,x)}\,\mathrm{d}\theta_1\,\mathrm{d}\theta_2,$$

where $\lambda>0$. For an interval $J$, write $x\cdot J=[xa,xb]$ when $J=[a,b]$, and let $|J|$ denote its length. Define

$$y_c=4\left(\frac{\lambda}{3}\right)^{1/2}.$$

Formal front-propagation conjecture. There exists an interval of trait values $J_0$ centered at $1$ such that, for every open interval $J\subset J_0$ centered at $1$, the density $f$ at large time is approximated by

$$f(t,x,\theta)=\begin{cases} \exp\left[-\frac{1}{4\lambda^2}\left(\theta-\lambda^{4/5}(6x^2)^{1/5}\right)^2+\underset{t\to\infty}{\mathcal O}(|J|^2)\right],&x\leq y_c t^{5/4},\quad \theta\in\lambda^{4/5}(6x^2)^{1/5}\cdot J,\\ \exp\left[\left(1-\left(\frac{x}{y_c t^{5/4}}\right)^{4/3}\right)t\right]\exp\left[-\frac{1}{4\lambda^2}\left(\theta-\left(\frac{3\lambda^2x^2}{2t}\right)^{1/3}\right)^2+\underset{t\to\infty}{\mathcal O}\left(|J|^2\frac{x^{8/3}}{t^{10/3}}\right)\right],&x\geq y_c t^{5/4},\quad \theta\in\left(\frac{3\lambda^2x^2}{2t}\right)^{1/3}\cdot J. \end{cases}$$

For $x\geq y_c t^{5/4}$, the prefactor of the trait distribution is $\exp\left[\left(1-\left(\frac{x}{y_c t^{5/4}}\right)^{1/3}\right)t\right]$, of the form $\exp[-c(x/(y_c t^{5/4}))t]$, where $c$ is positive and increasing on $(1,\infty)$. This is presented as a formal conjecture because the computations in the paper are only formal; it describes the evolving trait scale and the propagating front, including the decay of the population ahead of the front.