14 problems

Mass-growth conjecture. Analogous mass-growth and boundedness results should hold at least for the $\alpha$-stable operator (fractional Laplacian) $\mathcal L=(-\Delta)^{\alpha/2}$…

Let $n$ denote the bacterial population state and let $\theta,b7,d,b$ be the resource supply rate, dilution rate, death rate, and growth-rate coefficient, respectively. Assume that…

Let $\Omega$ and the given function $z$ be smooth as in Theorem, and let $u$ be a weak solution of the Dirichlet problem. The nonlocal Neumann datum $\mathcal N_s u$ is conjectured…

Singular-limit conjecture. The local model is the singular limit of the non-local model as $\mu\to\delta_{0^{+}}$.

The space-fractional Swift–Hohenberg equation is a non-local reaction-diffusion PDE whose critical modes arise near a bifurcation point. An amplitude equation is a reduced PDE gove…

Critical-profile conjecture. The rescaled function $u(t,e^{c_*t}\cdot)$ converges, as $t\to+\infty$, to a nontrivial limiting function.

Attractor conjecture. The energy functional could be used to prove that the attractor of the system is an unstable manifold of fixed points.

Steady-state convergence conjecture. The solution $\mathbf{u}$ converges towards a steady state.

Let $u$ be a solution of the half-harmonic gradient flow into $S^1$, with bubbling understood as concentration of the flow producing a bubble. Sire–Wei–Zheng conjecture. Bubbling m…

Gradient blow-up conjecture. Failure to include non-locality in the advection term, equivalently setting $K$ to be a Dirac delta function, will lead to gradient blow-up.

Non-linear feedback hypothesis. The failure of SDMs to make predictions in novel situations may be due, in part, to their failure to account for non-linear feedbacks in movement me…

For the rSV system with $obreakvarepsilon0$, let a solution develop a blow-up singularity at some finite time. Post-blow-up singularity conjecture. After the blow-up time, sol…

Consider the equation … with $0.25\leq\alpha<1$, and consider solutions that arise as limits from vanishing-viscosity approximations. Here $H$ denotes the Hilbert transform and…

Localized blowup conjecture. A solution blows up if and only if there exists some $x_0\in S^1$ such that