20 problems

Let $T$ and $x_{0}>0$ be fixed, and let $(\mu^{*}_{t})_{t\in[0,T]}$ be the deterministic solution considered in the allometric setting. Write $\mu^{*}_{t}=u_{t}(x)\,dx$ when it has…

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

Asymptotic bound conjecture. A stronger property is conjectured to hold: for any $n\geq 1$,

Let … be the square in the $z$-$w$ plane, and let … be the stability region, where $P_i(z,w)$ are the stability polynomials of the numerical method. Stability-region conjecture. Al…

Uniqueness conjecture. The limiting system above has a unique positive and bounded solution on $\mathbb{R}$.

Convergence conjecture. Under these assumptions,

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…

Kinetic integro-differential Krylov–Safonov conjecture. There exist constants $\alpha>0$ and $C$, depending only on the dimension, $\lambda$, and $\Lambda$, such that

Consider eigenproblems of the form … where $\alpha=2-2H\in(0,2)\setminus\{1\}$, $\mathbb{K}_{\alpha}$ is the integro-differential operator defined by … and the problem is equipped…

Let $\omega$ denote the state variable in the common-risk-factor component of the model, and let $\phi_\omega$ be the corresponding function. Let $a$ and $k$ be constants. Power-la…

Consider a Markovian claim $X=X^y$ driven by a jump diffusion $Y$, with drift $b$, volatility $sigma$, jump coefficients $gamma^{(i)}$, compensator measures $nu^i$, running loss…

General-kernel controllability conjecture. The same equivalence should remain true without the assumption $g(x,y)=g(x)$; namely, for kernels satisfying…

Reachability-to-zero conjecture. Every initial condition $\xi\in L^2(\Omega)$ can be controlled to hit the target

Heat-like controllability conjecture. The controllability properties of $(\mathrm{CGM})$ with $\alpha>0$ should be similar to those of the memoryless heat equation; in particular,…

Extension conjecture. The results on existence, uniqueness, and local attraction of traveling-wave profiles should extend to general BV initial data $z_{o}(u)$, provided that the t…

Mesh-grading conjecture. The condition

General comparison with measurable ingredients conjecture. Under these assumptions, the conclusion of the comparison proposition holds with $M^+_{CS}$ in place of $M^+_A$.

Let $u_s$ be a branch of $(2m-1)$-interface stable solutions of the steady-state equation with mean $s$, and let $\varepsilon_m^{b,s}$ be the value of $\varepsilon$ at which the sa…

Let $\varepsilon_m^0$ be the value at which the zero-mean $(2m-1)$-interface solution becomes stable, and let $\varepsilon_m^d$ be the value of $\varepsilon$ at which this branch b…

Fix $v\in(0,1)$ and consider nonnegative self-similar profiles $F$ with … where … and the mass condition is … so that $m_1=m_0/v$. The profile described in Theorem P.1 is nonnegati…