Vittoria Pierfelice
We prove a general dispersive estimate for a Schroedinger type equation on a product manifold, under the assumption that the equation restricted to each factor satisfies suitable dispersive estimates. Among the applications are the two-particle Schroedinger equations on Euclidean spaces, and the nonlinear Schroedinger equation on the product of two real hyperbolic spaces.
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Global solutions for 1D cubic defocusing dispersive equations, Part V: low regularity NLS
本文针对一维含广泛非线性项（含三次NLS）的散焦色散方程，利用密度-通量恒等式等方法，将全局适定性与色散界推广到负Sobolev空间$s > -1/2$的小数据情况，还可通过标度推广到大数据。
In this paper, we utilize the method in Dodson-Murphy [4] to establish the radial scattering result for the focusing nonlinear Schrödinger equation with inverse square potential $i\pa_tu-\la u=-|u|^{p-1}u$ in the energy space $H^1_a(\R^d)$ in dimensions $d\geq3$, which extends the result of [10,11] to higher dimensions cases but with radial initial data. The new ingredient is to establish the dispersive estimate for radial function and overcome the weak dispersive estimate when $a<0$.
We consider a nonlinear dispersive equation with a quasilinear quadratic term. We establish two results. First, we show that solutions to this equation with initial data of order $\mathcal{O}(\varepsilon)$ in Sobolev norms exist for a time span of order $\mathcal{O}(\varepsilon^{-2})$ for sufficiently small $\varepsilon$. Secondly, we derive the Nonlinear Schrödinger (NLS) equation as a formal approximation equation describing slow spatial and temporal modulations of the envelope of an underlying carrier wave, and justify this approximation with the help of error estimates in Sobolev norms bet…
On asymptotic stability in energy space of ground states of NLS in 2D
转置K.Yajima和T.Mizumachi的工作，证明了二维非线性薛定谔方程（NLS）基态线性化色散解的色散估计和光滑估计.作为应用，我们将文献中证明的非线性最小二乘系统基态渐近稳定性的结果推广到二维。
For nonlinear dispersive systems, the nonlinear Schrödinger (NLS) equation can usually be derived as a formal approximation equation describing slow spatial and temporal modulations of the envelope of a spatially and temporally oscillating underlying carrier wave. Here, we justify the NLS approximation for a whole class of quasilinear dispersive systems, which also includes toy models for the waterwave problem. This is the first time that this is done for systems, where a quasilinear quadratic term is allowed to effectively lose more than one derivative. With effective loss we here mean the lo…
Approximate Peregrine Solitons in Dispersive Nonlinear Wave Equations
本文将NLS近似的有效性从Sobolev空间推广到周期部分与实轴部分之和的函数空间，从而能利用Peregrine解在更复杂系统中研究怪波动力学。
We show, by the means of several examples, how we can use Gibbs measures to construct global solutions to dispersive equations at low regularity. The construction relies on the Prokhorov compactness theorem combined with the Skorokhod convergence theorem. To begin with, we consider the non linear Schrödinger equation (NLS) on the tri-dimensional sphere. Then we focus on the Benjamin-Ono equation and on the derivative nonlinear Schrödinger equation on the circle. Next, we construct a Gibbs measure and global solutions to the so-called periodic half-wave equation. Finally, we consider the cubic …
We prove the existence of a 2-parameter family of small quasi-periodic in time solutions of discrete nonlinear Schrödinger equation (DNLS). We further show that all small solutions of DNLS decouples to this quasi-periodic solution and dispersive wave.
In this paper, developing a new approach based on Fourier analysis methods for dispersive PDEs, we establish a low regularity NLS approximation for the one-dimensional cubic Klein-Gordon equation. Our main result includes energy class solutions which are formally asymptotically in $L^2(\mathbb{R})$. A precise rate of convergence is also obtained assuming more regularity.
In this paper, we revisit the infinite iteration scheme of normal form reductions, introduced by the first and second authors (with Z. Guo), in constructing solutions to nonlinear dispersive PDEs. Our main goal is to present a simplified approach to this method. More precisely, we study normal form reductions in an abstract form and reduce multilinear estimates of arbitrarily high degrees to successive applications of basic trilinear estimates. As an application, we prove unconditional well-posedness of canonical nonlinear dispersive equations on the real line. In particular, we implement this…
We continue our study of the cubic fractional NLS with very weak dispersion $α>1$ and data distributed according to the Gibbs measure. We construct the natural strong solutions for $α>α_0=\frac{31-\sqrt{233}}{14}\approx 1.124$ which is strictly smaller than $\frac{8}{7}$, the threshold beyond which the first nontrivial Picard iteration has no longer the Sobolev regularity needed for the deterministic well-posedness theory. This also improves our previous result in Sun-Tzvetkov \cite{Sun-Tz2}. We rely on recent ideas of Bringmann \cite{Bringmann} and Deng-Nahmod-Yue \cite{Deng2}. In particular …
Dispersive estimates for Schroedinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II
我们考虑非自伴算子的线性化NLS和证明色散边界的时间演化，而不假设的本质谱的边缘是经常性的。我们的方法不依赖于NLS的任何特定属性。相反，它是公理化的线性水平上，我们的结果是从四个假设（这当然是由NLS的动机）。这项工作是三维的。
Numerical evidence for singularity formation in defocusing fractional NLS in one space dimension
针对含分数次拉普拉斯算子和散焦非线性项的薛定谔型方程，通过小能量超临界s的数值模拟，为解中新型高振荡奇性形成提供数值证据。
We analyze the stationary and traveling wave solutions to a family of degenerate dispersive equations of KdV and NLS-type. In stark contrast to the standard soliton solutions for non-degenerate KdV and NLS equations, the degeneracy of the elliptic operators studied here allows for compactly supported steady or traveling states. As we work in $1$ dimension, ODE methods apply, however the models considered have formally conserved Hamiltonian, Mass and Momentum functionals, which allow for variational analysis as well.
Dispersive estimates for 1D matrix Schrödinger operators with threshold resonance
本文研究含阈值共振的一维非自伴矩阵 Schrödinger 算子，通过减去有限秩共振项证明加权色散与局部衰减率恢复至正则情形，并发现聚焦三次 NLS 扰动方程二次非线性的有利结构。
In this paper, we establish bilinear and gradient bilinear Strichartz estimates for Schrödinger operators in 2 dimensional compact manifolds with boundary. Using these estimates, we can infer the local well-posedness of cubic nonlinear Schrödinger equation in $H^s$ for every $s>{2/3}$ on such manifolds.
We consider the energy critical focusing NLS in R^3 and prove, for any $ν$ sufficiently small, the existence of radial finite energy solutions that as $t\to\infty$ behave as a sum of a dynamically rescaled ground state plus a radiation, the scaling law being of the form $t^ν$.
When a solution to the Cauchy problem for nonlinear dispersive equations is obtained by a fixed point argument using auxiliary function spaces, it is non-trivial to ensure uniqueness of solutions in a natural space such as the class of continuous curves in the data space. This property is called unconditional uniqueness, and proving it often requires some additional work. In the last decade, unconditional uniqueness has been shown for some canonical nonlinear dispersive equations by an integration-by-parts technique, which can be regarded as a variant of the (Poincaré-Dulac) normal form reduct…
In this paper we present a rigorous modulational stability theory for periodic traveling wave solutions to equations of nonlinear Schrödinger (NLS) type. We first argue that, for Hamiltonian dispersive equations with a non-singular symplectic form and $d$ conserved quantities (in addition to the Hamiltonian), one expects that generically ${\mathcal L}$, the linearization around a periodic traveling wave, will have a $2d$ dimensional generalized kernel, with a particular Jordan structure: The kernel of ${\mathcal L}$ is expected to be $d$ dimensional, the first generalized kernel is expected to…
A revision of "On asymptotic stability in energy space of ground states of NLS in 1D"
本文转置T.Mizumachi的工作，证明了一维非线性薛定谔方程（NLS）基态线性化色散解的光滑估计。作为应用，我们将Cuccagna和Mizumachi关于$\ge 3$维NLS基态的渐近稳定性的结果推广到一维
In this paper we study the asymptotic behavior of a quadratic Schrödinger equation with electromagnetic potentials. We prove that small solutions scatter. The proof builds on earlier work of the author for quadratic NLS with a non magnetic potential. The main novelty is the use of various smoothing estimates for the linear Schrödinger flow in place of boundedness of wave operators to deal with the loss of derivative. As a byproduct of the proof we obtain boundedness of the wave operator of the linear electromagnetic Schrödinger equation on an $L^2$ weighted space for small potentials, as well …
Local well-posedness of nonlinear dispersive equations on modulation spaces
利用时频分析的工具，我们得到了调制空间$M{p, 1}_{0,s}$中具有Cauchy数据的NLS，NLW和NLKG方程的局部适定性的一些改进结果.
In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state of Killip-Visan-Zhang by utilizing Dodson and Murphy's argument and the dispersive estimate established by Ivanovici and Lebeau, which avoids using the concentration compactness. We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary. Based on this effect and the interaction Morawetz estimate…
Consider the defocusing quintic nonlinear Schrödinger equation on $\mathbf{R}^3$ with initial data in the energy space. This problem is "energy-critical" in view of a certain scale-invariance, which is a main source of difficulty in the analysis of this equation. It is a nontrivial fact that all finite-energy solutions scatter to linear solutions. We show that this remains true under small compact deformations of the Euclidean metric. Our main new ingredient is a long-time microlocal weak dispersive estimate that accounts for the refocusing of geodesics.
Asymptotic stability of N-soliton states of NLS
聚焦非线性薛定谔方程具有特殊的非色散孤波型解，即孤子。在一定的谱假设下，我们证明了N个非碰撞孤子的线性组合解的存在性和渐近稳定性渐近稳定性.
We study the focusing intercritical NLS \begin{align}\label{abstract_nls} i\pt_t u+Δ_{x,y}u=-|u|^αu\tag{NLS} \end{align} on the semiperiodic waveguide manifold $\R^d_x\times \T_y$ with $d\geq 5$ and $α\in(\frac{4}{d},\frac{4}{d-1})$. In the case $d\leq 4$, with the aid of the semivirial vanishing theory \cite{Luo_inter}, the author was able to construct a sharp threshold, which being uniquely characterized by the ground state solutions, that sharply determines the bifurcation of global scattering and finite time blow-up solutions in dependence of the sign of the semivirial functional. As the d…
We prove existence and stability results for a two-parameter family of solitary-wave solutions to a system in which an equation of nonlinear Schrödinger type is coupled to an equation of Korteweg-de Vries type. Such systems model interactions between short and long dispersive waves. The results extend earlier results of Angulo, Albert and Angulo, and Chen. Our proof involves the characterization of solitary-wave solutions as minimizers of an energy functional subject to two constraints. To establish the precompactness of minimizing sequences via concentrated compactness, we establish the sub-a…
Dispersive decay for the Inter-critical nonlinear Schrödinger equation in $\mathbb{R}^3$
本文研究了三维空间中非临界非线性薛定谔方程的色散衰减问题，得出了初值在临界Sobolev空间中解的长期衰减估计。
A third-order dispersive flow for closed curves into Kähler manifolds
研究了Kähler流形中闭曲线的三阶色散方程的初值问题.该方程是描述涡丝运动的两球值系统的几何推广。利用几何分析和经典能量方法证明了局部存在性定理。
It is universally accepted that the cubic, nonlinear Schrodinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Kortewegde Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves and such a system has been studied over the last few decades. However, questions about the validity of the system in the study of water waves were raised in our previous work where we presented our analysis using the fifth…
We prove a Wong-Zakai theorem for the defocusing mass-critical stochastic nonlinear Schrödinger equation (NLS) on $\mathbb{R}$. The main ingredient are careful mixtures of bootstrapping arguments at both deterministic and stochastic levels. Several subtleties arising from the proof mark the difference between the dispersive case and corresponding situations in SDEs and parabolic stochastic PDEs, as well as the difference between the large-$n$ case and the limiting ($n=\infty$) case.
The KdV equation models the propagation of long waves in dispersive media, while the NLS equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A system that couples the two equations to model the interaction of long and short waves seems attractive and such a system has been studied over the last decades. We evaluate the validity of this system, discussing two main problems. First, only the system coupling the linear Schrödinger equation with KdV has been derived in the literature. Second, the time variables appearing in the equations are of a diff…
The initial value problem for a third-order dispersive flow into compact almost Hermitian manifolds
本文给出了在涡丝几何分析中产生的紧致几乎厄米特流形上开曲线的三阶色散发展方程的初值问题的局部存在性定理.这个方程导致了所谓的一阶导数损失，因为目标流形不应该是凯勒流形。我们克服了这个困难，通过使用规范变换的乘数上的拉回丛，以消除坏的一阶项本质上。
The bidirectional NLS approximation for the one-dimensional Euler-Poisson system
本文严格证明一维Euler-Poisson系统的解可由两个相向传播的NLS波包近似，扩展了单向结果，首次给出双向NLS近似的有效证明。
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2σq^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $H^{1,1}$ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main to…
A note on the NLS equation on Cartan-Hadamard manifolds with unbounded and vanishing potentials
本文研究了维数 $N \geq 3$ 的 Cartan-Hadamard 流形上的半线性 NLS 方程，在势函数允许无界和无穷远处衰减的条件下证明了非平凡解的存在性，并推广了 $\mathbb{R}^N$ 中的已有结果。
We discuss a short-time existence theorem of solutions to the initial value problem for a fourth-order dispersive flow for curves parametrized by the real line into a compact Kähler manifold. Our equations geometrically generalize a physical model describing the motion of a vortex filament or the continuum limit of the Heisenberg spin chain system. Our results are proved by using so-called the energy method. We introduce a bounded gauge transform on the pullback bundle, and make use of local smoothing effect of the dispersive flow a little.
Scattering, random phase and wave turbulence
本文针对实平面上三次二维薛定谔方程，利用色散效应和高斯截断周期函数，在不同时间尺度下研究时间演化算子向动力学算子的两种不同收敛机制，从而揭示共振流形正则性与散射现象的联系。
Stable manifolds for all monic supercritical NLS in one dimension
我们证明了只要扰动选择在适当空间的小超曲面上，一维空间中所有的超临界一次聚焦NLS都表现出扰动驻波的渐近稳定性.
Using suitable modified energies we study higher order Sobolev norms' growth in time for the nonlinear Schrödinger equation (NLS) on a generic $2d$ or $3d$ compact manifold. In $2d$ we extend earlier results that dealt only with cubic nonlinearities, and get polynomial in time bounds for any higher order nonlinearities. In $3d$, we prove that solutions to the cubic NLS grow at most exponentially, while for sub-cubic NLS we get polynomial bounds on the growth of the $H^2$-norm.
The aim of this paper is two fold. We show that if a complex function $F$ on $\C$ operates in the modulation spaces $M^{p,1}(\R^n)$ by composition, then $F$ is real analytic on $\R^2 \approx \C$. This answers negatively, the open question posed in [M. Ruzhansky, M. Sugimoto, B. Wang, Modulation Spaces and Nonlinear Evolution Equations, arXiv:1203.4651], regarding the general power type nonlinearity of the form $|u|^αu$. We also characterise the functions that operate in the modulation space $M^{1,1}(\R^n)$. The local well-posedness of the NLS, NLW and NLKG equations for the `real entire' nonli…
Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$
本文通过将斯特里查茨估计整合进伯克霍夫正规形，首次在色散背景下证明了圆上五次NLS方程对 $s>2/5$ 的微小不规则初值的长时间行为。
We establish space-time dispersive estimates for solutions to the wave equation on compact Riemannian manifolds with bounded sectional curvature, with the same exponents as for $C^\infty$ metrics. The estimates are for bounded time intervals, so by finite propagation velocity the results apply also on non-compact manifolds under appropriate uniform conditions. We assume a priori that in local coordinates the metric tensor components satisfy ${\rm g}_{ij}\in W^{1,p}$ for some $p>d$, which ensures that the curvature tensor is well defined in the weak sense, but this can be relaxed to any assumpt…
In this paper we study fractal solutions of linear and nonlinear dispersive PDE on the torus. In the first part we answer some open questions on the fractal solutions of linear Schrödinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV). In the second part, we study fractal solutions of the vortex filament equation and the associated Schrödinger map equation (SM). In particular, we construct global strong solutions of the SM in $H^s$ for $s>\frac3…
Resonant large deviations principle for the beating NLS equation
本文针对环面上拍频NLS方程，在随机初始数据仅支撑于两个Fourier模式的情形下证明了大偏差原理，并揭示共振能量交换会增大极端波出现的概率，表明非线性聚焦机制可使解的sup-范数概率分布尾部增厚。
The energy-critical defocusing nonlinear Schrödinger equation on 3-dimensional rectangular tori is considered. We prove that the global well-posedness result for the standard torus of Ionescu and Pausader extends to this class of manifolds, namely, for any initial data in $H^1$ the solution exists globally in time.
We study the long time behavior of small (in $l^2$) solutions of discrete nonlinear Schrödinger equations with potential. In particular, we are interested in the case that the corresponding discrete Schrödinger operator has exactly two eigenvalues. We show that under the nondegeneracy condition of Fermi Golden Rule, all small solutions decompose into a nonlinear bound state and dispersive wave. We further show the instability of excited states and generalized equipartition property.
Stable manifolds for an orbitally unstable NLS
我们构造了三维三次聚焦非线性最小二乘方程的孤立子周围的局部Lipschitz图，在该图上存在整体解，并且渐近稳定性和散射成立.
Global well-posedness for rough solutions of defocusing cubic NLS on three dimensional compact manifolds
本文借助多重线性特征函数估计、I-method及Strichartz估计，证明三维Zoll流形与球面积上低正则三次NLS的整体适定性，并给出$H^s$范数多项式界。
Global behaviour of nonlinear dispersive and wave equations
本文综述了非线性色散方程的大数据全局（和渐近）行为分析的最新进展，如非线性波（NLW），非线性薛定谔（NLS），波映射（WM），薛定谔映射（SM），广义Korteweg-de Vries（gKdV），Maxwell-Klein-Gordon（MKG）和Yang-Mills（YM）方程。将非线性分为\emph{subcritical}（在高频下比线性色散弱）、\emph{critical}（在所有频率下与线性色散相当）或\emph{supercritical}（在高频下比线性色散强）是这种分析的基础，而且最近的许多进展都是以存在临界守恒律的情况为中心的。我们讨论如何合成一个令人满意的临界（标度不变）的整体理论，开始的基本积木微扰分析，守恒定律，和单调性公式，但也纳入更先进的（和最近的）工具，如规范变换，浓度紧凑性，和感应能量。
Global solutions for 1D cubic defocusing dispersive equations: Part I
本文针对一维三次散焦NLS方程，提出一种新的非微扰方法，首次在小但非局部化的L^2初始数据下证明了全局适定性与散射，并得到全局L^6 Strichartz估计和双线性L^2界。
We consider the non-selfadjoint operator [\cH = [{array}{cc} -Δ+ μ-V_1 & -V_2 V_2 & Δ- μ+ V_1 {array}]] where $μ>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1(\R^2)\times L^1(\R^2)\to L^\infty(\R^2)\times L^\infty(\R^2)$ dispersive decay estimates for the evolution $e^{it\cH}P_{ac}$. We also obtain the following weighted estimate $$ \|w^{-1} e^{it\cH}P_{ac}f\|_{L^\infty(\R^2)\times L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)\times L^1(\R…
We consider soliton-like solutions of a variable coefficients, subcritical nonlinear Schrodinger equation (NLS). In a previous result, we proved the existence of a pure, global-in-time, generalized soliton with prescribed asymptotic as t tends to -infinity. In addition, we described the corresponding dynamics for all time, up to a small error term. In this paper we prove that the soliton is never pure as t tends to infinity. Indeed, we give a precise sharp lower bound on the defect induced by the potential on the soliton. This result shows the existence of nontrivial dispersive effects acting …
Exponential mixing for the randomly forced NLS equation
本文研究空间局域阻尼和随机噪声作用下受随机外力的非线性薛定谔方程不变测度的指数混合，利用指数渐近紧性与控制性质建立其遍历性，拓展了相关色散方程统计行为的系列研究。
We study the nonlinear Schrödinger equation posed on product spaces $\mathbf R^n\times \mathcal M^k$, for $n\geq 1$ and $k\geq1$, with $\mathcal M^k$ any $k$-dimensional compact Riemaniann manifold. The main results concern global well-posedness and scattering for small data solutions in non-isotropic Sobolev fractional spaces. In the particular case of $k=2$, $H^1$-scattering is also obtained.
Asymptotic stability of small solitons to 1D NLS with potential
研究了能量类中超临界一维非线性薛定谔方程$$ iu_t+u_{xx}=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}\times\mathbb{R}$,}$$的小孤波解的渐近稳定性. Gustafson-Nakanishi-Tsai \cite{GNT}在3维情况下使用端点Eschhartz估计研究了这个问题。为了证明孤立波的渐近稳定性，我们需要证明解的色散部分$v(t,x)$对于某些空间$X$属于$L^2_t(0,\infty;X)$。在1维情况下，这一性质并不仅仅来自于Eschenhartz估计。本文证明了Kato型的局部光滑效应在时间上是全局的，并将这个估计与Eschhartz估计相结合，得到了$\|(1+x^2)^{-3/4}v\|_{L^\infty_xL^2_t}<\infty$，它意味着孤立波的渐近稳定性.
We prove bilinear estimates for the Schrödinger equation on 3D domains, with Dirichlet boundary conditions. On non-trapping domains, they match the $\mathbb{R}^3$ case, while on bounded domains they match the generic boundary less manifold case. As an application, we obtain global well-posedness for the defocusing cubic NLS for data in $H^s_0(Ω)$, $1<s\leq 3$, with $Ω$ any bounded domain with smooth boundary.
Non-generic blow-up solutions for the critical focusing NLS in 1-d
本文讨论了临界一维非线性最小二乘问题伪共形爆破解的余维一稳定流形的存在性。通过Perelman和Merle，Raphael的工作，这些解的爆破速率与一般解的爆破速率相差甚远。然而，Bourgain，Wang和Perelman提出了非一般爆破解的余维单稳定流形的可能性。这篇论文建立了这种流形的存在性，尽管只是在可测范畴中。
Continuum limit for discrete NLS with memory effect
本文研究带长程相互作用和记忆效应的离散非线性薛定谔方程在网格尺寸趋于零时的连续极限，证明其解在L²中强收敛到连续NLS型方程的解并给出收敛速率，方法推广了Hong-Yang及Kenig-Ponce-Vega的经典估计框架。
We consider the focusing nonlinear Schrödinger equation $i u_t + Δu + |u|^{p-1}u=0$, $p>1,$ and the generalized Hartree equation $iv_t + Δv + (|x|^{-(N-γ)}\ast |v|^p)|v|^{p-2}u=0$, $p\geq2$, $γ<N$, in the mass-supercritical and energy-subcritical setting. With the initial data $u_0\in H^1(\mathbb{R}^N)$ the characterization of solutions behavior under the mass-energy threshold is known for the NLS case from the works of Holmer and Roudenko in the radial [16] and Duyckaerts, Holmer and Roudenko in the nonradial setting [10] and further generalizations (see [1,11,14]); for the generalized Hartre…
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