Related papers: Mass concentration for the $L^2$-critical Nonlinea…

In this paper, we show that any solution of the nonlinear Schr{{\"o}}dinger equation $iu\_t+\Delta u\pm|u|^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is…

We consider the $L^{2}$-critical quintic focusing nonlinear Schr\"odinger equation (NLS) on ${\bf R}$. It is well known that $H^{1}$ solutions of the aforementioned equation blow up in finite time. In higher dimensions, for $H^{1}$…

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require,…

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^{4/d} u$ for large spherically symmetric L^2_x(R^d) initial data in dimensions $d\geq 3$. In…

We consider the focusing mass-critical nonlinear Schr\"odinger equation and prove that blowup solutions to this equation with initial data in $H^s(\R^d)$, $s > s_0(d)$ and $d\geq 3$, concentrate at least the mass of the ground state at the…

We consider finite time blowup solutions of the $L^2$-critical cubic focusing nonlinear Schr\"odinger equation on $\R^2$. Such functions, when in $H^1$, are known to concentrate a fixed $L^2$-mass (the mass of the ground state) at the point…

In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} -\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm…

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 observe a link between the window size of mass concentration and the rate of explosion of the Strichartz norm by revisiting Bourgain's mass concentration for the L^2-critical nonlinear Schrodinger equations.

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 study the non-scattering $L^{2}$ solution $u$ to the radial mass critical nonlinear Schr\"odinger equation with mass just above the ground state, and show that there exists a time sequence $\{t_{n}\}_{n}$, such that $u(t_{n})$ weakly…

In this paper, we prove a refined version of a compactness lemma and we use it to establish mass-concentration for the focusing nonlinear Schr\"{o}dinger equation with an inverse-square potential.

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…

We consider the $L^2$-critical NLS with inverse-square potential $$ i \partial_t u +\Delta u + c|x|^{-2} u = -|u|^{\frac{4}{d}} u, \quad u(0) = u_0, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, $$ where $d\geq 3$ and $c\ne 0$ satisfies…

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 study the concentration phenomenon for solutions of the fractional nonlinear Schr\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation…

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 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 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 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…