相关论文: Mass-energy Scattering Criterion For Double Power …

We investigate the scattering theory for the nonlinear Schr\"{o}dinger equation $i \partial_{t}u+ \Delta u+\lambda|u|^\alpha u=0$ in $\Sigma=H^{1}(\mathbb{R}^{d})\cap L^{2}(|x|^{2};dx)$. We show that scattering states $u^{\pm}$ exist in…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means…

In this paper, we consider the $L_x^2$-scattering of defocusing mass sub-critical nonlinear Schr\"odinger equations with low weighted initial condition. It is known that the scattering holds with $\mathcal{F} H^1$-data, while the continuity…

We consider time global behavior of solutions to the focusing mass-subcritical NLS equation in weighted $L^2$ space. We prove that there exists a threshold solution such that (i) it does not scatter; (ii) with respect to a certain…

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^\alpha u = 0\qtq{on}\R\times\R^N, \] with $\alpha=\tfrac{4-2b}{N-2}$, $N=\{3,4,5\}$ and $0<b\leq…

In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schr\"odinger equation with a defocusing energy-subcritical perturbation term under a ground state energy threshold in four spatial dimension.…

We consider the focusing $5$d Hartree equation, which is $L^2$-supercritical, with finite energy initial data, and investigate the solutions at the mass-energy threshold. We establish the existence of special solutions following the work of…

In this paper, we simplify the proof of M. Hamano in \cite{Hamano2018}, scattering theory of the solution to \eqref{NLS system}, by using the method from B. Dodson and J. Murphy in \cite{Dodson2018}. Firstly, we establish a criterion to…

In this paper, we study the defocusing energy-critical nonlinear Schr\"odinger equations $$ i\partial_t u + \Delta u = |u|^{\frac{4}{d-2}} u. $$ When $d=3,4$, we prove the almost sure scattering for the equations with non-radial data in…

In this article, we consider the dynamics of the energy-critical quadratic nonlinear Schr\"odinger system $\[ \left\{ \begin{aligned} & i u^1_t + \kappa_1 \Delta u^1 = -\overline{u^2}u^3, \\ & i u^2_t + \kappa_2 \Delta u^2 =…

We consider the linear wave equation with the time-dependent scale-invariant damping and mass. We also treat the corresponding equation with the energy critical nonlinearity. Our aim is to show that the solution scatters to a modified…

In this paper, we continue the study in \cite{MiaoWZ:NLS:3d Combined} to show the scattering and blow-up result of the solution for the nonlinear Schr\"{o}dinger equation with the energy below the threshold $m$ in the energy space…

This note studies the asymptotic behavior of global solutions to the fourth-order Schr\"odinger equation $$i\dot u+\Delta^2 u+F(x,u)=0 .$$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing…

We study the Cauchy problem for nonlinear Schr\"odinger equations with attractive inverse-power potentials. By using variational arguments, we first determine a sharp threshold of global well-posedness and blow-up for the equation in the…

The direct and inverse scattering problems on the full line are analyzed for a first-order system of ordinary linear differential equations associated with the derivative nonlinear Schr\"odinger equation and related equations. The system…

We consider the long-time dynamics of focusing energy-critical Schr\"odinger equation perturbed by the $\dot{H}^\frac{1}{2}$-critical nonlinearity and with inverse-square potential(CNLS$_a$) in dimensions $d\in\{3,4,5\}$…

In this paper, we consider a class of the defocusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u - |x|^{-b} |u|^\alpha u = 0, \quad u(0)=u_0 \in H^1, \] with $b, \alpha>0$. We firstly study the decaying…

In this article we investigate the nature of the functions, including important double power terms which arise naturally in considering typical nonlinear Schroedinger equations.

We investigate the nonlinear Schr\"{o}dinger equation $iu_{t}+\Delta u+|u|^{p-1}u=0$ with $1+\frac{4}{N}<p<1+\frac{4}{N-2}$ (when $N=1, 2$, $1+\frac{4}{N}<p<\infty$) in energy space $H^1$ and study the divergent property of…

We study the stochastic nonlinear Schroedinger equations with linear multiplicative noise, particularly in the defocusing mass-critical and energy-critical cases. For general initial data, we prove the global existence and uniqueness of…