Related papers: Scattering for radial bi-harmonic Hartree equation…

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…

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…

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…

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…

The purpose of this work is to study the $3D$ energy-critical inhomogeneous generalized Hartree equation $$ i\pa_tu+\Delta u+|x|^{-b}(I_\alpha\ast|\cdot|^{-b}|u|^{p})|u|^{p-2}u=0,\;\ x\in\R^3, $$ where $p=3+\alpha-2b$. We establish global…

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 energy-critical nonlinear Schr\"odinger equation of fourth order $iu_t+\Delta^2 u=|u|^\frac{8}{d-4}u$. We prove that if a maximal-lifespan radial solution $u: I\times\Bbb R^d\to\mathbb{C}$ obeys…

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 $\mathbb{R}\times \mathbb{R}^d$, $d\geq 5$.

In this paper, we study the global well-posedness and scattering for the wave equation with a cubic convolution $\partial_{t}^2u-\Delta u=\pm(|x|^{-3}\ast|u|^2)u$ in dimensions $d\geq4$. We prove that if the radial solution $u$ with…

We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in dimension $d \in \{3,4,5\}$ and prove the exterior scattering of solutions if $3\leq d \leq 5$ and $1+6/d<p<1+4/(d-2)$. More…

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…

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 study the asymptotic behavior of solutions to the defocusing mass-subcritical Hartree NLS $iu_t + \Delta u = F(u) = (|x|^{-\gamma}*|u|^2)u$ on $\mathbb{R}^d$, $d\geq 2$, $\frac{4}{3} < \gamma < 2$. We show that the scattering problem…

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…

In this work we consider a wide range of energy critical wave equation in 3-dimensional space with radial data. We are interested in exterior scattering phenomenon, in which the asymptotic behaviour of a solutions $u$ to the non-linear wave…

We consider the inhomogeneous nonlinear Schr\"odinger equation $$ i u_t +\Delta u+|x|^{-b}|u|^\alpha u = 0, $$ where $\frac{4-2b}{N}<\alpha<\frac{4-2b}{N-2}$ (when $N=2$, $\frac{4-2b}{N}<\alpha<\infty$) and $0<b<\min\{N/3,1\}$. For a radial…

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…

We investigate the defocusing inhomogeneous nonlinear Schr\"odinger equation $$ i \partial_tu + \Delta u = |x|^{-b} \left({\rm e}^{\alpha|u|^2} - 1- \alpha |u|^2 \right) u, \quad u(0)=u_0, \quad x \in \mathbb{R}^2, $$ with $0<b<1$ and…

We consider the fractional Hartree equation in the $L^2$-supercritical case, and we find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}<M[Q]^{\frac{s-s_c}{s_c}}E[Q]$…

We describe the asymptotic behavior as time goes to infinity of solutions of the 2 dimensional corotational wave map system and of solutions to the 4 dimensional, radially symmetric Yang-Mills equation, in the critical energy space, with…