12 problems

Consider the discrete coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels, with initial data $\rho_0$ satisfying the paper's initi…

Classification conjecture. For $\beta=(d+\alpha)/(d-\alpha)-1$, the functions $u_{c,A}$ constitute all bounded solutions of the stationary equation, while for…

Stationary-solution and stability conjecture. When $m>m_*$, the model should be identical to the case $m<1/2$ in terms of the number of stationary solutions and their stability.

Let $d_1$ be the first critical strip width, and consider solutions with Neumann boundary conditions satisfying the symmetries under discussion. Uniqueness conjecture. This uniquen…

Let $\rho$ and $\eta$ be stationary solutions supported on $[-b,b]$ and $[-c,c]$, respectively, with $0<b<c$, satisfying … … for constants $C_1$ and $C_2$. Energy-minimizer conject…

Let $\Gamma$ be a star graph with $N\geq 3$ edges, let $p>1$, let $\beta\ne 0$, and let $\omega>0$. Let $\phi_\beta$ denote the symmetric profile, and let equation (d) be the stati…

Let $g:[0,1]\to[0,1]$ be measurable, and consider the stationary solution obtained from the hydrodynamic limit in either of the regimes $\gamma\in(0,1)$ and $\theta>0$, or…

Consider the nonlocal dispersal equation in the exterior of a compact obstacle $K\subset\mathbb{R}^N$, without assuming that $K$ is convex, and let $u$ be the entire solution const…

Nonexistence conjecture. There are no nodal solutions with more than one sign change, namely

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…

For $j>0$ and mass $M$, consider the stationary profile … These profiles form a one-parameter family of stationary solutions indexed by the conserved mass. Uniqueness conjecture fo…

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…