86 problems

Let $u$ solve the one-dimensional $L^2$-critical nonlinear Schrödinger equation with repulsive delta potential $u<0$, with initial data $u0in H^1(mathbb{R})$ satisfying … where…

Global well-posedness and scattering conjecture. Then $u$ is global and scatters, and

Let $u$ solve the radial, defocusing, cubic wave equation … on $\mathbf{R}^{3}$ with initial data $(u_0,u_1)\in \dot{H}^{1/2}(\mathbf{R}^{3})\times\dot{H}^{-1/2}(\mathbf{R}^{3})$,…

Global well-posedness and scattering conjecture. For every $E\in(0,\infty)$ and every smooth initial datum $\vec{u}_0$ with $E_{ls}(\vec{u}_0)\leq E$, there exists a global solutio…

Consider the defocusing cubic nonlinear Schrödinger initial value problem … with initial data $\phi(x,0)=\phi_0(x)\in H^s(\mathbb{R}^3)$, where $H^s(\mathbb{R}^3)$ is the usual inh…

Let $Q$ be the ground state solution of the focusing cubic nonlinear Schrödinger equation, and consider the deterministic Zakharov system in the energy space…

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…

Global well-posedness and scattering conjecture. Global well-posedness and scattering hold for $(1.1)$ when $0<s_c<1$. The mass-critical case $s_c=0$ and energy-critical case…

Global scattering theorem. In the four sign regimes, the following conclusions hold: if $\mu_1=\mu_2=-1$, then $u$ is global and scatters; if $\mu_1=1$ and $\mu_2=-1$, then $u$ is…

Small-data asymptotic-freedom conjecture. If

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…

Translated rigidity conjecture. Theorem 5.1 remains true if

Spherically symmetric scattering conjecture. In the defocusing case ($\mu=+1$) we have $m_{0,\operatorname{rad}}=+\infty$, while in the focusing case ($\mu=-1$) we have…

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…

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 $B_\delta^-(0)$…

Non-trapping obstacle conjecture. The local energy of solutions should decay for any non-trapping obstacle, and consequently such solutions should scatter.

Quadratic-phase conjecture. For the combined nonlinear Schrödinger equation, if $b>0$, the solution with initial condition $v_0$ exists globally and scatters, whereas if $b<0$ and…

Let $d\geq1$, $1+\frac{2}{d}<q<p<1+\frac{4}{d}$, and let $\rho_{0,E}>0$ be the threshold mass for the existence of ground states. Let $\rho^{\star}<\rho_{0,E}$ be the small-mass sc…

Let $u_0$ be initial data for the one-dimensional fourth-order mixed-dispersion nonlinear Schrödinger equation, with power $\alpha<8$ and parameter satisfying $a<\sqrt{b}$. Let…

The Klein–Gordon–Zakharov system is considered in spacetime dimension $\mathbb{R}^{1+d}$, with scattering behavior as described in Theorem … may be a phenomenon specific to the two…

Consider the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential and damping function, under the hypotheses of Theorem 1 in the source, which gives…

Consider the inhomogeneous nonlinear Schrödinger equation with inverse-square potential in the focusing case, with parameters $N$, $\lambda$, and $b$, and let…

Critical-norm global well-posedness and scattering conjecture. Under these assumptions, global well-posedness and scattering should hold under a priori control of the critical norm…

Critical-norm global well-posedness and scattering conjecture. If $s_c>0$, then $u$ is global and scatters as $t\to\pm\infty$.

Scattering-threshold conjecture. The quantity $\|Q\|_{L^2}$ is the scattering threshold for the focusing equation.