23 problems

Let $u$ solve the Fisher–KPP equation … The initial data are understood in the take-over setting considered for this equation, with traveling-wave states $\phi(-\infty)=1$ and…

Randomized-Gumbel conjecture. The maximum particle, centered by $m(t)$, converges in distribution to a randomized Gumbel.

Let $\overline{\phi}\in \overline{K}(\mathbb{R})$ be an admissible kernel perturbation supported in $[-1/2,1/2]$, with $||\overline{\phi}||_1^m$ small. Consider parameters…

Exact spreading-speed conjecture. The free boundary $h(t)$ satisfies

Let $u(x,t)$ solve the Neumann problem … with … and initial data $u(x,0)=u_0(x)$, where $u_0(x)>0$ for some $x$. Long-time convergence conjecture. … The claim describes the predict…

Consider the one-dimensional Dirichlet problem … with the Dirichlet boundary conditions and initial data specified in the paper, and let $u$ denote its solution. Large-population w…

Convergence conjecture. The solution $u(t,x)$ converges to the traveling wave $U$ as $t\to+\infty$ in the moving frame with speed $c_{het}$. This concerns the long-time asymptotic…

Marginal stability conjecture. Strongly localized initial data propagate with the linear spreading speed $c_*(\delta)$, where solutions of the linearized equation generically grow…

Consider the nonlocal Fisher–KPP equation … with nonnegative initial value $U_0$. Its dampening term is $-u^\alpha\int_{\mathbb{R}^n}u\,dx$. Nonlocal dampening conjecture. The damp…

Let $u\colon\mathbb{R}\to\mathbb{R}$ be the population density, let $a^-$ be the competition kernel, and let $\varkappa^+$, $\varkappa^-$, and $m$ denote the corresponding reproduc…

Bounded front-width conjecture. For $b7<1$, almost surely,

For a measure $\mu$ as in Theorem, suppose that there is $\zeta\in S^{d-1}$ such that $\operatorname{supp}(\mu)\subseteq \mathcal{W}_{0,\zeta}$, but $\operatorname{supp}(\mu)$ is n…

Refined Fisher–KPP expansion conjecture. One has

Level-difference conjecture. One has

Ebert–van Saarloos conjecture. If $u_0(x)$ decays “fast enough”, then

Universal first correction conjecture. One has

Fast-diffusion exponential propagation conjecture. For every $\sigma<\sigma_{\ast}$,

Full-classification conjecture. This classification holds for every solution $0<u(t,x)<1$ of the one-dimensional equation, without assuming the backward-decay condition.

Full decay-characterization conjecture. For any solution $0<u<1$, the limit $\lambda$ exists in $[0,\sqrt{f'(0)}]$, and $u$ is a transition front connecting $0$ and $1$ if and only…

Asymptotic-speed and profile conjecture. Any such transition front has asymptotic past and future speeds $c_\pm$ satisfying

Critical-speed traveling-front conjecture. Transition fronts with global mean speed

The nonlocal Fisher-KPP equation has the constant stationary state $u\equiv 1$. Supercritical Hopf-type bifurcation conjecture. The state $u\equiv 1$ may undergo a supercritical Ho…

Generalization conjecture. The conclusions might apply to these more general versions of the Fisher–KPP equation whenever $g(0)=0$.