22 problems

Consider two open selection problems: selecting harmonic wavenumbers through fronts, and finding selected fronts that leave hexagonal patterns in the wake. Secondary-front explanat…

Consider modulated invasion in which the leading edge, the front interface, or the state in the wake may be oscillatory or temporally constant. Oscillatory invasion conjecture. The…

Let $u_\mathrm{pl}(\xi,\tau)$ be a modulated pulled pattern-forming front satisfying the stated pinched-double-root, periodic-wake, exponential-asymptotic, marginal-Floquet-stabili…

Let $u_\mathrm{ps}(\xi,\tau)$ be a modulated pushed pattern-forming front, namely a temporally modulated pushed front leaving a pattern-forming wake. Selection of modulated pushed…

Let $u_\mathrm{pl}$ be a rigid pulled pattern-forming front, meaning a rigid front with marginal spectral stability that leaves behind a diffusively stable periodic pattern and has…

Let $u_\mathrm{tr}$ be a pushed-to-pulled transition front. A pushed-to-pulled transition front is a front at the boundary between pushed and pulled invasion, characterized here by…

Consider the competitive Lotka–Volterra system … … with $a_1<1<a_2$, and let the pulled-regime condition be … where $M=\max\{1,2(1-a_1)\}$. Fix $a_1=\frac12$ and $\sigma=1$. Hosono…

Small-noise speed-limit conjecture. The limit

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…

Let $K(\theta)$ be the directional spreading speed and define the Wulff shape by … Let $\hat{\mathscr{F}}$ be the convex hull of the Frank diagram, let…

Consider the bidomain Allen--Cahn equation and its planar and zigzag fronts. The Frank diagram may have a portion inside its convex hull where its curvature is positive, while plan…

Let $\theta$ be the direction of a zigzag front, and let $\theta_{\mathrm{m}}(t)$ and $\theta_{\mathrm{p}}(t)$ determine the two angles forming a peak, namely…

Consider a boundary case at the transition between pulled and pushed fronts, where the front propagates with the linear spreading speed $c_*$. Let $eta_*$ denote the spatial decay…

A one-sided invasion process connects a pointwise unstable state ahead of the front to a selected state in its wake. A front is called marginally stable when it lies at the thresho…

Let $U_{\alpha,1}$ denote the value function associated with the exponential ansatz discussed in the strategy section, and let $t$ denote time. Asymptotic value-function conjecture…

Let $u_\omega$ be the solution from Definition, under the hypotheses of Theorem (i) but without its limitation on the dimension $d$, and let $L_{u_\omega,\eta}(t)$ denote the trans…

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

Spreading-rate conjecture. The same rate of spreading holds for initial data which are growing sufficiently fast at infinity.

Let $L>0$ and consider the $L$-branching Brownian motion ($L$-BBM), in which particles are killed when they are more than distance $L$ behind the rightmost particle. Let $c$ denote…

Front-location conjecture. Any lack of physical realism in the original model is relatively insignificant for long-term behaviour; more concretely, the front location behaves simil…

Consider the kinetic reaction-transport equation with KPP-type growth, and let $0$ denote its unstable steady state. The Méndez–Campos–Gómez-Portillo conjecture. As for the KPP equ…

Consider the infection model with diffusion rates $0=D_R<D_B$ and particle density parameter $\rho$. The asymptotic velocity of the infection front is the limiting speed of the fro…