12 problems

Conjecture star. For $n\leq8$, $\Gamma$ is the graph of a function

Let $e\in\mathbb{S}^{n-1}$ and let $c_e^\star$ denote the critical speed in direction $-e$. A planar pulsating traveling wave solution has speed $c$, propagates in direction $-e$,…

Lambda-profile conjecture. $\Lambda$-shaped profiles should appear in $\Omega_{(0,1)}(u)$, so the inclusion in Theorem and Corollary might be strict.

For each direction $e\in\mathbb{S}^{N-1}$, let $c_e$ denote an admissible speed of a pulsating front. A function of $e$ is sign-unchanging if it is continuous and does not change s…

Monotonicity conjecture. The quantity $|d_T|$ is nonincreasing as a function of $T>0$. Consequently, $d_T$ has a fixed sign independent of $T>0$, equal to the sign of

Let $\mathbfcal{A}$ denote the set of speed functions admissible for spatially periodic bistable reaction-diffusion equations, with each speed function assigning a propagation spee…

Periodic nonsingularity conjecture. In a periodic domain, there exist nontrivial solutions in which no singularity forms.

The periodic domain $\Omega_\varepsilon$ is a three-dimensional grating of equi-spaced parallel pipes with constant cross sections, oriented along the three orthogonal directions…

Consider two-species Volterra–Lotka competition systems with nonlocal dispersal in a time-periodic and spatially inhomogeneous medium. The species with slower diffusion is said to…

Let $L$, $d$, $\alpha$, $k$, $f_{1}$ and $f_{2}$ be the parameters and functions defining the periodic competition problem, and let a bistable pulsating front solution mean a pulsa…

Let $a(x)$ and $f(x,u)$ be general diffusion and reaction coefficients satisfying the bistable assumptions and the assumptions on the spatial dependence stated in the source, and c…

Sommerfeld–Bethe conjecture. In dimensions $d=2$ and $d=3$, the number of gaps is finite.