14 problems

Consider a nonlinear diffusion model on the sphere whose interaction kernel has an eigenfunction expansion … where the $Y_l$ are eigenfunctions of the Laplace–Beltrami operator, an…

Uniqueness conjecture. The self-similar profile with this asymptotic behavior is unique.

Let $u$ and $v$ be solutions for which the relative mass is conserved, meaning that the integral of their difference is invariant in time. For $p>1$, consider the corresponding rel…

Nonlinear-diffusion metastability conjecture. The same results should hold in the nonlinear diffusion case with $Q(0)=0$.

Infinite-speed and waiting-time conjecture. A discontinuity develops with infinite initial speed whenever

Let $d$ be the spatial dimension and let $\gamma>0$ be the parameter in the Fokker–Planck equation. The regime considered in the paper is … The preceding discussion concerns the co…

Slow-motion conjecture. The long-time asymptotics of any solution in $X_+^2$ should correspond to slow motion along $\mathcal{E}$, described to leading order by either of these dif…

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…

Consider the diffusion-absorption model with parameter $\lambda>0$ and discontinuous Heaviside diffusivity having jump $\epsilon$. Let $\alpha$ denote the time-decay parameter in t…

Consider a globally defined bounded solution $(u,g,h)$ of the free-boundary problem described above, in the case where … where $v$ is an evenly decreasing positive solution of…

Existence of the asymptotic shift limit. The limit

Let $\Omega$ be a finite-volume open convex domain with $C^\infty$ boundary, let $f\in L^2(\Omega)$, and let $\rho_{(\varepsilon)}(x,t)$ solve the regularized problem … where…

The long-time Gaussian asymptotics conjecture. If $a+2b+3c-3>0$ and the initial condition $f$ is small enough, then there is a constant $A$, usually depending on $f$, $g$, $\mu$,…

Consider the Cauchy problem … where $q>1$, and let $q_c=(N+2)/(N+1)$. Nonlinear-reaction dominance conjecture. For every non-zero initial datum $u_0$, the nonlinear reaction term…