相关论文: Ground state mass concentration in the L^2-critica…

We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schr\"odinger equation \[ i\partial_t u -(-\Delta)^s u = -|u|^\alpha u, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times…

We study the focusing $L^2$-critical and supercritical stochastic nonlinear Schr\"odinger equation subject to additive or multiplicative noise. We investigate global or long time behavior of solutions in $H^1$, which would correspond to…

In this paper we study dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schr\"odinger equation. We firstly establish the profile decomposition of…

We consider the $L^2$-critical nonlinear Schrodinger equation with an inhomogeneous damping term. We prove that there exists an initial data such that the corresponding solution is global in $H^1(R^d)$ and we give the minimal time of the…

We study the focusing 3d cubic NLS equation with H^1 data at the mass-energy threshold, namely, when M[u_0]E[u_0] = M[Q]E[Q]. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering…

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation in $H^1(\mathbb{R}^3)$, \begin{equation} i\partial_t u + \Delta u + |x|^{-b}|u|^{2}u=0,{equation} where $0 < b <\tfrac{1}{2}$. Previous works have established a…

We consider the following nonlinear Schr\"{o}dinger equation with an inverse potential: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2\sigma}}u=0 \] in $\mathbb{R}^N$. From the classical argument, the…

We are concerned with the multi-bubble blow-up solutions to rough nonlinear Schr\"odinger equations in the focusing mass-critical case. In both dimensions one and two, we construct the finite time multi-bubble solutions, which concentrate…

We consider the focusing nonlinear Schr\"odinger equation with the critical inverse square potential. We give the first proof of the uniqueness of the ground state solution. Consequently, we obtain a sharp Hardy-Gagliardo-Nirenberg…

We consider the $L^2$-critical nonlinear Schr\"odinger equation (NLS) with the delta potential $$i\partial_tu +\partial^2_x u + \mu \delta u +|u|^{4}u=0, \, \, t\in \R, \, x\in \R , $$ where $ \mu \in \R$, and $\delta$ is the Dirac delta…

We consider the following nonlinear Schr\"{o}dinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+\Delta u+|u|^\frac{4}{3}u+\mu\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where…

This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schr\"odinger equation of power type with data in a critical homogeneous weighted $L^2$ space. We give a sharp sufficient condition for…

In this paper, we will consider the $L^2$-critical fractional Schr\"odinger equation $iu_t-|D|^{\beta}u+|u|^{2\beta}u=0$ with initial data $u_0\in H^{\beta/2}(\mathbb{R})$ and $\beta$ close to $2$. We will show that the solution blows up in…

We consider the nonlinear Schr\"odinger equation \[ u_t = i \Delta u + | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical…

In this paper, we consider the dynamics of the solution to the mass critical focusing Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$. The main difficulties are the equation is \emph{not}…

We characterize the dynamics of the finite time blow up solutions with minimal mass for the focusing mass critical Hartree equation with $H^1(\mathbb{R}^4)$ data and $L^2(\mathbb{R}^4)$ data, where we make use of the refined…

In this article we obtain new scattering and blow-up solutions for intercritical focusing nonlinear Schr\"{o}dinger equations (NLS) above the ground state mass-energy threshold. The main focus of this article is the establishment of some…

We consider the following nonlinear Schr\"{o}dinger equation with double power nonlinearity \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u+|u|^{p-1}u=0,\quad 1<p<1+\frac{4}{N} \] in $\mathbb{R}^N$. For $N=1,2,3$, Le…

We consider the inhomogeneous nonlinear Schr\"odiger equation (INLS) in $\mathbb{R}^N$ $$i \partial u_t + \Delta u + |x|^{-b} |u|^{2\sigma}u = 0,$$ and show the $L^2$-norm concentration for the finite time blow-up solutions in the…

We consider the nonlinear Schr\"odinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer…