Related papers: Scattering for the generalized Hartree equation wi…

By $I$-method, the interaction Morawetz estimate, long time Strichartz estimate and local smoothing effect of Schr\"odinger operator, we show global well-posedness and scattering for the defocusing Hartree equation $$\left\{…

We establish global existence, scattering for radial solutions to the energy-critical focusing Hartree equation with energy and $\dot{H}^1$ norm less than those of the ground state in $\mathbb{R}\times \mathbb{R}^d$, $d\geq 5$.

We study the theory of scattering in the energy space for the Hartree equation in space dimension n>2. Using the method of Morawetz and Strauss, we prove in particular asymptotic completeness for radial nonnegative nonincreasing potentials…

We consider the defocusing, $\dot{H}^1$-critical Hartree equation for the radial data in all dimensions $(n\geq 5)$. We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we…

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Hartree equation $iu_t+\Delta u=\pm(|x|^{-2}*|u|^2)u$ for large spherically symmetric $L^2_x(\Bbb{R}^d)$ initial data; in the focusing case we…

We study the generalized Hartree equation, which is a nonlinear Schr\"odinger-type equation with a nonlocal potential $iu_t + \Delta u + (|x|^{-b} \ast |u|^p)|u|^{p-2}u=0, x \in \mathbb{R}^N$.We establish the local well-posedness at the…

In this paper, we study the long-time behavior of global solutions to the Schr\"odinger-Choquard equation $$i\partial_tu+\Delta u=-(I_\alpha\ast|\cdot|^b|u|^{p})|\cdot|^b|u|^{p-2}u.$$ Inspired by Murphy, who gave a simple proof of…

We consider the defocusing supercritical generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u-\partial_x(u^{k+1})=0$, where $k>4$ is an even integer number. We show that if the initial data $u_0$ belongs to $H^1$ then…

In this paper we study the small data scattering of Hartree type semirelativistic equation in space dimension $3$. The Hartree type nonlinearity is $[V * |u|^2]u$ and the potential $V$ which generalizes the Yukawa has some growth condition.…

In this paper, we consider the 3d cubic focusing inhomogeneous nonlinear Schr\"{o}dinger equation with a potential $$ iu_{t}+\Delta u-Vu+|x|^{-b}|u|^{2}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{3}}, $$ where $0<b<1$. We first establish…

In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u-V(x)u…

In this paper, we study the focusing and defocusing energy--subcritical, nonlinear wave equation in $\mathbb{R}^{1+d}$ with radial initial data for $d = 4,5$. We prove that if a solution remains bounded in the critical space on its interval…

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 consider a class of $L^2$-supercritical inhomogeneous nonlinear Schr\"odinger equations with potential in three dimensions \[ i\partial_t u + \Delta u - V u = \pm |x|^{-b} |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^3, \]…

The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schr\"odinger equation $$ i u_t +\Delta u+|x|^{-b}|u|^2 u = 0, $$ where $0<b<1/2$. Let $Q$ be the ground state solution of $-Q+\Delta Q+ |x|^{-b}|Q|^{2}Q=0$ and…

We mainly consider the focusing biharmonic Schr\"odinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(\Delta^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{N}}, u(0,…

In this paper, we consider the following three dimensional defocusing cubic nonlinear Schr\"odinger equation (NLS) with partial harmonic potential \begin{equation*}\tag{NLS} i\partial_t u + \left(\Delta_{\mathbb{R}^3 }-x^2 \right) u = |u|^2…

In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{tt}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the…

In this paper, we prove the decay and scattering in the energy space for nonlinear Schr\"odinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + \Delta u - V(x)u + \lambda |u|^{p - 1}u = 0$. We will prove decay…

We investigate the well-posedness in the generalized Hartree equation $iu_t + \Delta u + (|x|^{-(N-\gamma)} \ast |u|^p)|u|^{p-2}u=0$, $x \in \mathbb{R}^N$, $0<\gamma<N$, for low powers of nonlinearity, $p<2$. We establish the local…