104 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 $L^2$-supercritical generalized Korteweg–de Vries equation in the odd-power case, and the convergence around spatial zero asserted in Theorems … . Odd-power convergenc…

Consider the defocusing nonlinear wave equation … with dimension $d\geq 3$, and let the initial data be radially symmetric. The scaling-critical initial-data space is…

De Giorgi's conjecture. There exists a limit

The Adkins–Nappi model is a semilinear modification of the wave map equation describing a repulsive interaction involving meson fields. By finite energy smooth initial data, we mea…

Consider the focusing energy-critical wave equation … and the Talenti–Aubin solution $W(r)=(1+r^2/3)^{-1/2}$. Let $\mathcal M$ be the small codimension-one manifold around $W$ whos…

Conjecture on the transmission exponent. The exponent in this asymptotic is exactly

Critical-energy conjecture. The critical energy is exactly

Let $B_\delta(0)$ be a neighborhood of the origin in the data space, and let $\Sigma$ be the stable manifold of the stationary solution, so that $B_\delta^+(0)$ and $B_\delta^-(0)$…

Large-measure periodic-solution conjecture. For $M=0$, periodic solutions should exist for a large-measure set of values of the amplitudes.

Let $\mathcal G$ be a compact Lie group with Lie algebra $\mathbf g$, let $s_0=\frac{n-2}{2}$ be the scaling index, and consider the Yang–Mills equation … for a connection with cur…

Let $(M,g)$ be a complete Riemannian manifold, and consider wave maps from $\mathbb R^2\times\mathbb R$ into $(M,g)$. The critical initial-data space is $H^1\times L^2$. Wave maps…

Let $n\geq 4$, let $0\leq s\leq \frac12$, and consider the power nonlinear wave equation … The initial data lie in $H^s\times H^{s-1}$. Local well-posedness conjecture. The equatio…

For a semilinear wave equation, let $H^s \times H^{s-1}$ denote the Sobolev data space, and let $s_0$ be the scaling index determined by the equation's scaling. Below-scaling ill-p…

The systems under consideration are basic field theories with critical well-posedness exponent $s_c$, meaning that the $dot H^{s_c} \times dot H^{s_c-1}$ norm of the initial data i…

Consider the dispersive Hamiltonian normal form … where $k_{n+1,n}=d_{n+1,n}+ic_{n+1,n}$, and suppose the first relevant resonance occurs at a higher harmonic…

Consider all solutions of the five-dimensional energy-critical wave equation satisfying the fixed-center, comparable-scale concentrating multi-bubble asymptotic assumptions for $K$…

Fix spatial centers $z_1,\ldots,z_K\in\mathbb{R}^5$ and let $Eq(F)$ be the algebraic equilibrium set governing the rescaled modulation dynamics of the corresponding multi-bubble so…

Let $(u,\partial_tu)$ be a $K$-bubble solution of the five-dimensional energy-critical wave equation in the fixed-center, comparable-scale concentrating regime. Let…

Let $K\geq 2$, let $z_1,\ldots,z_K\in\mathbb{R}^5$ be pairwise distinct, and consider a global forward-in-time solution of the five-dimensional focusing energy-critical wave equati…

Let $u$ be a solution of the focusing energy-critical wave equation in five spatial dimensions, with lifespan endpoint that may be finite or infinite. A soliton is a rescaled and t…

Bizon–Rostworowski's resonant-instability conjecture. The instability is driven by a mechanism of resonant interactions of spherically symmetric modes, leading to weakly turbulent…

Let $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$ and let $\mathcal{C}$ be the backward light cone. Let $\phi$ solve the relevant wave-map or semilinear equation…

Let $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$ and let $\mathcal{C}$ be the backward light cone. The symbols $\phi$, $r$, $\nu$, $d$, $p$, and $k$ have the me…

Let $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$, and let $\phi$ be a solution arising from smooth initial data in the stated Type I self-similar stability resu…