26 problems

For problem, consider positive, radially symmetric solutions with the setting described above. Fila–King's conjecture. Infinite-time blow-up occurs only in dimensions $3$ and $4$.…

Upper-bound conjecture for the extrapolation range. The extrapolation range is limited from above by $q_+(\mathcal{H})$.

Let $u_0:\mathbb{R}\to\mathbb{R}$ be an initial condition for the free boundary problem (FBP), with initial support $(L_0,R_0)$, parameter $p$, and solution $u$ having boundaries…

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 $p>1$ and let denote the diffusive Hamilton–Jacobi problem studied in the source. An entire solution is a solution defined for all $t\in\mathbb{R}$; it is nonstationary if it d…

Let $\Sigma$ be the lateral boundary of a parabolic domain, given by the graph of a $\operatorname{Lip}(1,1/2)$ function. Hunt's conjecture. Caloric measure should be mutually abso…

The two-dimensional Keller–Segel system is known to admit finite-time blowup solutions when the initial density has total mass greater than $8\pi$ and finite second moment; existin…

Interpolation embedding conjecture. Such an interpolation result should hold for the case of a weak Lipschitz domain.

Let $N\geq 3$ and consider the logarithmic Keller–Segel system obtained when $\gamma(s)=s^{-k}$, with parameter $k$ and critical value $k_c=\frac{N}{N-2}$. The threshold conjecture…

Let $(M,g)$ be a complete, $n$-dimensional Riemannian manifold with bounded geometry. An ancient, positive solution is a positive solution $u$ of … that is defined for all past tim…

Let $u$ be the solution of the Cauchy–Dirichlet problem under the assumptions of the preceding theorem, and let $T$ be its first singular time. Complete blowup conjecture. The blow…

Vanishing-scale transfer-spectrum conjecture. For at least the considered test case, this independence result should transfer to

Reduced-oversampling conjecture. Less oversampling should already be sufficient to extract the significant modes.

Domain-independent local-space conjecture. In general, it should be possible to construct suitable local reduced spaces without knowledge of the global computational domain…

Boundary-lifting regularity conjecture. Less regularity of $g_N$ should still be sufficient to infer

Let Problem be the fast diffusion equation with critical exponent and Dirichlet boundary conditions studied in the paper, and let a bubble tower solution mean a solution exhibiting…

Extension conjecture. Theorem holds even without this second restriction, although the constant $C$ may then depend on $T$ as well.

Indirect controllability conjecture. As in the constant-coefficient case, either the first line of or the condition $g_{21}\neq 0$ in $(a,b)\times\mathcal O$ is sufficient for indi…

Controllability conjecture. The system of two coupled linear parabolic equations above should be controllable with the single distributed control $\mathds{1}_{\omega}v$.

Let $\rho$ be a solution to the porous medium equation on $[-1,0] \times B_2(0)$. Assume that there is a nonempty open cone $C$ such that $\rho(t,x) \leq \rho(t,y)$ whenever…

Let $u$ solve the classical nonlinear heat equation … For $N\geq 3$, set $p_S=(N+2)/(N-2)$, while $p_S=\infty$ for $N=1,2$. Optimal Liouville conjecture. The Liouville property sho…

Existence and uniqueness claim. The problem has a unique viscosity solution $u$.

Let $\Omega$ be the domain, let $p$ be the exponent, let $m\geq 1$, and let ${\bf v}(x,t)$ be the solution under consideration. Let $\Upsilon_p$ denote the exponential rescaling ra…

Critical-growth non-existence conjecture. If

Let $\gamma:[0,T]\to\mathbb{R}^{N+1}$ be a $C^1$ curve, write $\gamma(\tau)=(x(\tau),t(\tau))$, and assume $\gamma'(\tau)\neq 0$. For $\tau_0\in(0,T)$, let…