6 problems

Energetic stability threshold conjecture. There is $\omega_* \in (\omega_L,1)$ and $\Omega_* \in (0,1)$ such that the even wave satisfying (even-wave) is energetically stable for…

Scattering conjecture for critical gKdV. There exist functions

Let $m>3$, let $u_0\in X\cap H^{1/2}(\mathbb R)$ be sufficiently localized, where $X$ is the corresponding local well-posedness space for the given $m$, and let $Q$ be the ground s…

Let $u_0\in H^{1/2}(\mathbb R)$ be sufficiently localized, let $Q$ be the ground state solution of the equation defining $Q$ with $m=3$, and let $u(x,t)$ be the mBO time evolution…

Let $u$ solve the focusing, mass-critical generalized KdV equation $(1.1)$ with initial data $u(0,x)=u_0\in L^2(\mathbb{R})$, and let … be the ground state, which satisfies…

Suppose that $u \in C^0([0,T);H^{1/2}(\mathbb{R}^3))$ solves the boson star equation with $m\geqslant 0$ and blows up at a finite time $0<T<+\infty$. Let $u^*\in L^2(\mathbb{R}^3)$…