14 problems

Let $\rho$ be a solution of the two-dimensional pseudo-gravity model with diffusion coefficient $D$, and let $M_0$ denote the conserved mass. The entropy estimate indicates dissipa…

Numerical-error conjecture. The numerical stationary state appearing to exceed $\rho_E$ for values such as $m=2.1$ and $m=2.5$ is a numerical error, as the stationary state moves a…

Let $G$ be a non-negative integrable kernel, and consider the aggregation-diffusion models and associated free boundary problems discussed in the paper, with energies whose…

Conjecture on the small-diffusion limit. As $D\rightarrow 0$, the time to decay of a two-peaked solution tends to infinity, so that two-peaked solutions become stable.

Conjecture on local energy minima and transient states. Local energy minima of the approximate system are qualitatively similar to the asymptotic patterns observed in the aggregati…

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…

Saddle-node bifurcation conjecture. The red points are all saddle-node bifurcations and correspond to discontinuous phase transitions for $m\geq 2$.

A related problem with linear diffusion concerns the stability of local minimizers of the zero-diffusion energy when a small noise parameter is introduced. The phenomenon under dis…

Multiple large-time behavior conjecture. For $<\varepsilon<1$, the equation features multiple behavior for large times: it has more than one attractor as $t\to+\infty$.

Let $\chi>0$, and let $\rho_{HLS,\lambda^*,c^*}\in{\mathcal Y}$ be the density whose parameters $(\lambda^*,c^*)$ are given by the stated parameter relation. Instability conjecture…

Let $\chi>0$, $k\in(-N,0)$, and $m=2N/(2N+k)$ in dimension $N\geq1$. Assume that the initial datum $\rho_0\in{\mathcal Y}$ is radially symmetric. Global-existence-versus-blow-up co…

Let $\chi>0$, $k\in(-N,0)$, and $m=2N/(2N+k)$ in dimension $N\geq1$. Assume that the initial datum $\rho_0\in{\mathcal Y}$ is radially symmetric, and let $\rho_{HLS,\lambda,c^*}$ d…

Let ${\bar\rho}_{k,1}$ be the stationary profile and let ${\mathcal J}_\infty$ denote the limiting value of the second-moment-related quantity in the two-case analysis. Admissibili…

Let $m>1$ and, for each support size $R>0$, let $(\varepsilon(R),R,\rho_R)$ denote the corresponding stationary solution triple, with $\varepsilon(R)$ the diffusion coefficient and…