66 problems

Consider the radial semilinear wave equation … where $f(u)=-3u(1-u^2)$, and let … Here $U_0$ is the self-similar solution written in terms of the similarity variable $\rho=r/(T-t)$…

Consider the complex Ginzburg–Landau equation with parameter $\theta=\pi/2$, namely the nonlinear Schrödinger equation. A solution is unbounded if its norm becomes arbitrarily larg…

Scattering conjecture. In the defocusing case ($\mu=+1$) we have $m_0=+\infty$, while in the focusing case ($\mu=-1$) we have $m_0=M(Q)$. This predicts global well-posedness and sc…

Generic-dynamics conjecture. Generic perturbations of an excited soliton should lead either to blowup, or to collapse to a less energetic soliton (or to the vacuum state), plus rad…

Stable-manifold threshold conjecture. The codimension-one stable manifold of the solution $f_1$ plays the role of the threshold of singularity formation for a large set of initial…

Consider the critical dimension $d=4$ radial Yang–Mills equation … where $\Delta_{(d-2)}=\partial_r^2+\frac{d-3}{r}\partial_r$, and let … be the static instanton profile. Let $t_*$…

Consider the one-dimensional cluster model with dimension parameter $d$ and interaction exponent $\alpha$, and extend it to higher dimensions $d>1$. Critical-value conjecture. In h…

Delayed-blowup conjecture. The corresponding solution of either managed model does not blow up in $[0,T]$; thus, blowup can be delayed by dispersion or nonlinearity management.

Let a system of hyperbolic conservation laws have totally linearly degenerate characteristics, and let its initial data belong to $H^s(\mathbb R^n)$ with $s\geq 1+\frac{n}{2}$. Maj…

Consider the three-dimensional Keller–Segel equation and the explicit self-similar solution … where … and $Q$ satisfies … Here, $\Lambda$ denotes the scaling operator, and a radial…

Extremal threshold conjecture. For the extremal case $\sigma=1$, there is still a threshold separating dispersion to zero from finite-time ODE-type blowup in the white regions.

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…

Let $k\geq 2$, let $\varepsilon_k^*=\frac{k}{k+1}$, and consider the two-dimensional quasilinear wave equation … . Suppose that … and … for $\xi_0=-1$ and…

Stable-manifold conjecture. Such initial data are globally well-posed under the nonlinear Schrödinger dynamics if and only if $u_0\notin\mathcal{W}^s(0)$.

Cho et al.'s global well-posedness conjecture. The equation is globally well-posed for every real initial datum, whether small or large.

Integral criterion conjecture. Small-data global existence holds if

Chen–Reissig's conjecture. If $C_{\mathrm{Str}}>0$, then the solution blows up at finite time for some initial data; if $C_{\mathrm{Str}}=0$, then small-data global existence holds…

The co-rotational Skyrme model is considered in spatial dimension $d=5$, with the profile from the paper's equation (ProfileU) and the scaling properties of the full equation. Blow…

For the Yang–Mills system in the energy-supercritical range $d\geq 5$, consider the higher-dimensional Bizoń-Biernat self-similar solution and generic large-data evolutions in the…

Consider the Yang–Mills system in temporal gauge in dimension $d\geq 5$, restricted to the equivariant solution class, and let … where $\phi$ is the Bizoń-Biernat self-similar prof…

Threshold-role conjecture. The solution $u_T^*$ plays an important role in the study of threshold phenomena for Equation $$ .

For equivariant wave maps into the sphere in spatial dimension $d\geq 7$, let a corotational self-similar profile mean a solution of the self-similar wave-map equation in the corot…

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…

Continuity conjecture for blowup. Blowup phenomena should be continuous with respect to the parameter $\tau\to 0$, both for ${\rm (TM)}\to{\rm (NLH)}$ and for…

Biernat–Bizoń conjecture. Based on numerical simulations, Biernat and Bizoń conjectured that $A_T$ is the generic blowup profile within the equivariant solution class. This conject…