Global existence and blowup for Choquard equations with an inverse-square potential

Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential

We consider a class of the focusing nonlinear Schrödinger equation with inverse-square potential \[ i\partial_t u + Δu -c|x|^{-2}u = - |u|^αu, \quad u(0)=u_0 \in H^1, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $\frac{4}{d}\leq α\leq \frac{4}{d-2}$ and $c\ne 0$ satisfies $c>-λ(d):=-\left(\frac{d-2}{2}\right)^2$. In the mass-critical case $α=\frac{4}{d}$, we prove the global existence and blowup below ground states for the equation with $d\geq 3$ and $c>-λ(d)$. In the mass and energy intercritical case $\frac{4}{d}<α<\frac{4}{d-2}$, we prove the global existence and blowu…

The focusing cubic NLS with inverse-square potential in three space dimensions

We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS. We also prove failure of uniform space-time bounds at the threshold.

Scattering in $H^1$ for the intercritical NLS with an inverse-square potential

We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.

脉冲耦合振荡器平均场系统的经典解：长时间渐近性与爆破

Classical solutions of a mean field system for pulse-coupled oscillators: long time asymptotics versus blowup

本文通过逆分布函数重构脉冲耦合振荡器平均场系统，揭示距离的收缩/扩张，从而阐明长时间行为、稳态、收敛速率及爆破，并解释存在性障碍与放电率估计。

The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse-square potential

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse-square potential \[iu_{t} +Δu-c|x|^{-2}u=λ|x|^{-b} |u|^{σ} u,\; u(0)=u_{0} \in H^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $λ=\pm1$, $0<b<2$, $σ=\frac{4-2b}{d-2}$ and $c>-c(d):=-\left(\frac{d-2}{2}\right)^{2}$. We first prove the local well-posedness as well as small data global well-posedness and scattering in $H^{1}$ for $c>-\frac{(d+2-2b)^{2}-4}{(d+2-2b)^{2}}c(d)$ and $0<b<\frac{4}{d}$, by using the contraction mapping principle based on the Strich…