相关论文: On collapsing ring blow up solutions to the mass s…

We consider the energy super critical nonlinear Schr\"odinger equation $$i\pa_tu+\Delta u+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical…

We consider the minimal mass $m_0$ required for solutions to the mass-critical nonlinear Schr\"odinger (NLS) equation $iu_t + \Delta u = \mu |u|^{4/d} u$ to blow up. If $m_0$ is finite, we show that there exists a minimal-mass solution…

In this paper, we prove a universal upper bound on the blowup rate of a focusing nonlinear Schr\"odinger equation with an angular momentum under a trapping harmonic potential, assuming that the initial data is radially symmetric in the…

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

We consider the $L^2$ critical inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^N$ $$ i \partial_t u +\Delta u +|x|^{-b} |u|^{\frac{4-2b}{N}}u = 0, $$ where $N\geq 1$ and $0<b<2$. We prove that if $u_0\in…

We consider the nonlinear Schr\"odinger equation $iu_t=-\Delta u-|u|^{p-1}u$ in dimension $N\geq 3$ in the $L^2$ super critical range $1+\frac{4}{N}<p<\frac{N+2}{N-2}$. The corresponding scaling invariant space is $\dot{H}^{s_c}$ with…

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 intercritical inhomogeneous nonlinear Schr\"odinger equation. For the radial symmetry initial data, we construct the ring blow-up solutions and obtain blow-up speed. This result implies that the upper bound on…

The blowup is studied for the nonlinear Schr\"{o}dinger equation $iu_{t}+\Delta u+ |u|^{p-1}u=0$ with $p$ is odd and $p\ge 1+\frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative…

In this paper, we investigate a universal blow-up bound for the focusing mass-critical nonlinear Schr\"odinger equation for general initial data in $L^2(\mathbb R^d)$, extending previous knowledge for mass near the ground-state threshold…

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…

In this paper, we consider the following equation: \[ i\frac{\partial u}{\partial t}+\Delta u+g(x)|u|^{\frac{4}{N}}u-Wu=0. \] We construct a critical-mass solution that blows up at a finite time and describe the behaviour of the solution in…

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 inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^N$ \begin{align}\label{inls} i \partial_t u +\Delta u +V(x)|u|^{\frac{4-2b}{N}}u = 0, \end{align} where $V(x) = k(x)|x|^{-b}$, with $b>0$. Under suitable…

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…

In this paper we consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation \begin{align}\label{inls} i \partial_t u +\Delta u +|x|^{-b} |u|^{2\sigma}u = 0, \,\,\, x \in \mathbb{R}^N \end{align} with $N\geq 3$. We focus on the…

This paper is dedicated to the blow-up solution for the divergence Schr\"{o}dinger equations with inhomogeneous nonlinearity (dINLS for short) \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)=-|x|^c|u|^pu,\quad\quad u(x,0)=u_0(x),\] where…

In the first part of this paper, we investigate the sharp threshold of blow-up and global existence for the focusing nonlinear Schr\"{o}dinger equation with combined nonlinearities of mass-critical and mass-subcritical power-type.…

This paper is devoted to the analysis of blow-up solutions for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities \[ iu_{t}+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u. \] When $p_1=\frac{4}{N}$ and…

We consider the two dimensional $L^2$ critical nonlinear Schr\"odinger equation $i\pa_tu+\Delta u+u|u|^2=0$. In the pioneering work \cite{BW}, Bourgain and Wang have constructed smooth solutions which blow up in finite time $T<+\infty$ with…