20 problems

Let the stochastic time-fractional slow or fast diffusion equation be equipped with multiplicative noise, and let “our condition” refer to the sufficient parameter condition establ…

Let $a$ denote the damping parameter and let $\beta$ satisfy … The $L^2$ norm measures the size of the difference between the damped and inviscid solutions. Sharp uniform vanishing…

Fractional fast diffusion optimal-rate conjecture. There exists a constant $C>0$ such that

Consider the time-spectral method for a time-fractional diffusion problem of order $\alpha$, with $0<\alpha<1$. In the homogeneous case, let $f=0$ and $u_0\in\dot H^2(\Omega)$; in…

Let $u$ solve the half-order time-fractional diffusion equation with source $g(x,t)=f(x)R(x,t)$ in $Q$, under the zero initial condition and homogeneous Dirichlet boundary conditio…

Let $\gamma\in(0,2)\setminus\{1\}$, let $\hat\kappa>0$, let $g:[0,1]\to[0,1]$ be measurable, and let $\rho^{\hat\kappa}$ be a weak solution of the regional fractional reaction-diff…

Let $\gamma\in(0,2)$ and let $\theta>0$ be sufficiently small. Consider the hydrodynamic equation for the boundary-driven super-diffusive symmetric exclusion process, with $\kappa$…

Graded-mesh reconstruction-error conjecture. In general, provided $r\ge2$,

Superconvergence conjecture. For $r\ge2$, there is a constant $C$ such that

Let $f$ satisfy (F1) with $q\geq 1$, suppose that $f'(u)F(u)\leq q$ for all sufficiently large $u>0$, and let $\phi\geq 0$. Set … If … then (S1E1) has a local-in-time solution. Exi…

Fractional-operator inverse conjecture. The nonequilibrium steady-state energy correlations are given, up to a constant factor $\nu$, by the inverse of the fractional Laplacian:

Let $N\in\mathbb N$, let $\kappa\in[1,N]$ be an integer, let $s>0$, $\beta\geq 0$, and let $p$ satisfy … Let $\mathcal S$ be a closed set contained in an affine plane of codimensio…

Let $s>0$, $\beta\geq 0$, $p>1$, and let $K\subset\mathbb R^N$ be compact. Denote by $\operatorname{cap}^{\mathbb R^N}_{\frac{2s(1+\beta)}{p},p'}(K)$ the corresponding Bessel capac…

Consider the fractional diffusion inverse problem in which the source has the form $a(t)\chi(D)$, where $a(t)$ is an unknown time-dependent amplitude and $D$ is an unknown spatial…

Consider the chemotaxis model with critical diffusion $\Lambda u$ and no logistic term, and let the mass refer to the conserved total mass of the solution. Threshold-mass conjectur…

Let $\alpha\in(0,2)$ and let $\varpi_{p,\ell}^{(\alpha)}$ denote the coefficients used in the high-order numerical discretizations, with $p\in\{3,4,5\}$ and $\ell$ the coefficient…

Numerical-stability conjecture. The asymmetric case is not only more complicated with respect to the design and implementation of numerical algorithms but also with respect to nume…

Refined approximation conjecture. The law of $\lambda^{\frac12}P_{\cdot/\lambda}$ satisfies

Let $\mathcal{Q}(g,f)$ denote the Boltzmann collision operator acting on a function $f$ with collision background $g$, and let $(-\Delta_v)^s$ be the flat fractional Laplacian in t…

Let $u_M(x,t)$ be the unique nonnegative self-similar solution of … with initial data $u(x,0)=M\delta_0$, where $\alpha\in(1,2)$, $M>0$, and $b\in\mathbb{R}^d$ is fixed. Let…