57 problems

Explicit-form conjecture. An extremal for the general case $p\neq 2$ has a similar explicit form.

Energetic criterion for blow-up for $p>2$. If $p>2$, then $u(t)$ should blow up in finite time.

K2 residue conjectures. The following hold:

Pole and residue conjecture. The function $K_2(t,\pm1,\pm1)$ has simple poles precisely at $t=2$ and $t=-2m$, $m\in\mathbb{N}$; moreover, when $x=y\neq\pm1$, $K_2(t,x,x)$ has simpl…

Nonvanishing conjecture. The function $K_1(t,\varphi,\varphi)$ has simple poles exactly at $t=\frac12-m$, $m\in\mathbb{N}_0$; equivalently,

Let $u_0$ be initial data on $4\mathbb{R}^n4$, and let $4\alpha4$ and $p$ satisfy the hypotheses of problem (44). In particular, assume … The initial data are required to satisfy ……

Singular-measure existence conjecture. The same existence results should continue to hold when $\nu$ is singular with respect to $\mathcal{L}$.

Non-local modulus of continuity conjecture. Then $\varphi(\cdot,t)$ is a modulus of continuity for $u(\cdot,t)$ on $\Omega\times[0,\infty)$. This proposes that the higher-dimension…

Let $\Omega$ be the bounded domain in the nonlocal Poisson problem, let $u$ solve that problem, let $f\in L^2(\Omega)$, and let $\beta$ be the kernel exponent. Continuity conjectur…

Consider the nonlocal Poisson equation on a bounded domain $\Omega$, with forcing term $f$ and singular kernel exponent $\beta>3$. Continuity conjecture. Even if $f$ is discontinuo…

Boundary regularity conjecture. At least in smooth domains $\Omega\subset\mathbb{R}^n$, one has

Masoero–Raimondo string-equation conjecture. Under the nonvanishing assumption, $U(T,X)$ is a global solution of this equation.

Positivity conjecture for $K$. The conclusion of Lemma is also true when $sp=n$.

A priori-bounds conjecture. The conclusions of these two theorems still hold without the condition $p<4s+1$.

Nonlocal transverse-stability conjecture. Nonlocal effects can suppress the transverse instability of dark solitons.

Let $J$ be a kernel that decays sufficiently fast, and let $F_\varepsilon$ denote the equilibrium of the corresponding nonlocal Fokker–Planck equation. Poisson-like tail conjecture…

Consider the dissipative CCF equation and solutions obtained as vanishing-viscosity approximations, with dissipation exponent $\gamma\in[\frac12,1)$. Silvestre–Vicol's Hölder regul…

Let $p\neq 2$, and consider extremal functions for the $p$-fractional Sobolev inequality, with $C$ a constant. Up to translation and dilation, the extremal-function conjecture. Suc…

Convergence conjecture. For $(a,D)\in\mathcal{H}_1$, the solution to (DIVP) approaches the temporally stable unique $r$-peak steady state as $t\to\infty$ uniformly for $x\in[0,a]$.

Let $K$ be a jumping kernel satisfying the upper bound, Poincaré, Sobolev, and uniform jump comparability assumptions appearing in the full parabolic Harnack inequality theorem. Re…

Let $\Omega\subseteq\mathbb R^N$ be a convex set, and let $K$ be a kernel for which the associated operator $L_K$ is 1-regular. A classical solution refers to a sufficiently regula…

Let $\Omega\subseteq\mathbb R^N$ be an open bounded Lipschitz set satisfying a uniform exterior ball condition. Let $u$ be a classical solution of … with $\alpha\in(0,1)$ as in the…

Consider solutions $v\colon\mathbb{R}\to\mathbb{R}$ of the fractional equation … that satisfy … Here $P$ is the nonlocal operator … with the kernel $K$ defined by the fractional co…

Let $\varphi$ be a highest periodic traveling-wave solution of the Whitham equation, with a cusp at $x=0$ and crest height $\mu$. In the normalization of the Whitham equation, cons…

Assume that $f_a(u)=u(1-u)(u-a)$ with $0<a<1/2$. Let $\mathcal{K}$ satisfy the stated hypotheses with normalized second moment … Let $\mathcal{R}_j$, $j=1,\ldots,5$, be the regions…