Related papers: The energy-critical inhomogeneous generalized Hart…

In this work, we consider the focusing generalized inhomogeneous Hartree equation with potential \[ i u_t + \Delta u - V(x)u + \left(I_{\gamma} * |x|^{-b}|u|^{p}\right)|x|^{-b}|u|^{p-2}u = 0, \] where $0<\gamma<3$ and…

This paper studies the asymptotic behavior of global solutions to the generalized Hartree equation $$i\dot u+\Delta u+(I_\alpha *|\cdot|^b|u|^p)|x|^b|u|^{p-2}u=0 .$$ Indeed, using a new approach due to \cite{dm}, one proves the scattering…

We consider the focusing generalized Hartree equation in $H^1(\R^3)$ with a potential, \begin{equation*} iu_t + \Delta u - V(x)u + (I_\gamma \ast |u|^p )|u|^{p-2} u=0, \end{equation*} where $I_\gamma = \frac{1}{|x|^{3-\gamma}}$, $p \geq 2$…

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…

We consider the focusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation \[ iu_t + \Delta u = -|x|^{-b}|u|^{\alpha}u \] where $n \geq 3$, $0<b<\min(2, n/2)$, and $\alpha=(4-2b)/(n-2)$. We prove the global well-posedness and…

This manuscript proves the energy scattering of global solutions to a repulsive fourth-order generalized Hartree equation with non-radial data in the inter-critical regime. This work uses a new approach due to Dodson-Murphy [4] and extends…

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 investigate the focusing inhomogeneous nonlinear biharmonic Schr\"odinger equation \[ i\partial_t u + \Delta^2 u - |x|^{-b}|u|^p u = 0 \quad \text{on } \mathbb{R} \times \mathbb{R}^N, \] in the energy-critical regime, $p = \frac{8 -…

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 investigate the following inhomogeneous nonlinear Schr\"odinger equation in the radial regime, featuring a focusing energy-critical nonlinearity and a defocusing perturbation: $$ i\partial_t u +\Delta u =|x|^{-a} |u|^{p-2} u - |x|^{-b}…

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 consider the focusing nonlinear Schr\"odinger equation $i u_t + \Delta u + |u|^{p-1}u=0$, $p>1,$ and the generalized Hartree equation $iv_t + \Delta v + (|x|^{-(N-\gamma)}\ast |v|^p)|v|^{p-2}u=0$, $p\geq2$, $\gamma<N$, in the…

The paper deals with the defocusing case of the energy subcritical non-linear wave equation in $R^3$. We assume the initial data is in the space $\dot{H}^s \times \dot{H}^{s-1}$ and radial. If $s=1$, this is the energy space and the…

We study the well-posedness for the inhomogeneous Hartree equation $i\partial_t u + \Delta u = \lambda(I_\alpha \ast |\cdot|^{-b}|u|^p)|x|^{-b}|u|^{p-2}u$ in $H^s$, $s\ge0$. Until recently, its well-posedness theory has been intensively…

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $3\leq p<5$. We generalize inward/outward energy theory and weighted…

This note studies the asymptotic behavior of global solutions to the fourth-order generalized Hartree equation $$i\dot u+\Delta^2 u\pm(I_\alpha*|u|^p)|u|^{p-2}u=0.$$ Indeed, for both attractive and repulsive sign, the scattering is obtained…

This work studies the Cauchy problem for the energy-critical inhomogeneous Hartree equation with inverse square potential $$i\partial_t u-\mathcal K_\lambda u=\pm |x|^{-\tau}|u|^{p-2}(I_\alpha *|\cdot|^{-\tau}|u|^p)u, \quad \mathcal…

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

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…

We consider the focussing energy-critical inhomogeneous nonlinear Schr\"odinger equation: $$ iu_t + \Delta u + g|u|^2u = 0, u(0)= \varphi \in \dot{H}^1,\;\; 0 \le g_i \le |x|g \le g_s.$$ On the road map of Kenig-Merle \cite{km} we show the…