Tarek Saanouni, Hanene Hezzi
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 previous work [14] by removing the spherically symmetric assumption on the data.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
This paper studies the asymptotic behavior of global solutions to the generalized Hartree equation $$i\dot u+Δu+(I_α*|\cdot|^b|u|^p)|x|^b|u|^{p-2}u=0 .$$ Indeed, using a new approach due to \cite{dm}, one proves the scattering of the above inhomogeneous Choquard equation in the mass-super-critical and energy sub-critical regimes with radial setting.
This paper studies the inhomogeneous fractional Schödinger equation $$i\dot u-(-Δ)^s u=\pm(I_α*|\cdot|^b|u|^p)|x|^b|u|^{p-2}u.$$ In the mass super-critical and energy sub-critical regimes, using a Gagliardo-Nirenberg adapted to the above problem, the standing waves give a sharp threshold of global existence versus finite time blow-up of solutions.
Sharp weighted Strichartz estimates and critical inhomogeneous Hartree equations
本文针对非齐次Hartree方程在Sobolev临界指标下初值问题未解决的现状，通过建立所有可能的带奇异权的$L^p$ Strichartz估计，证明了临界情形下解的局部存在性。
Remarks on the well-posedness of the energy-critical inhomogeneous Hartree equation
本文研究空间维度三维及以上能量临界非齐次Hartree方程，通过建立新参数范围内的局部适定性，填补了先前未解决的参数区域，特别在维度5和6中完全解决了剩余缺口。
In this paper, we settle the problem for time-dependent Hartree equation with inhomogeneous boundary condition in a bounded Lipschitz domain in $\mathbb{R}^{N}$. A global existence result is derived.
We study a class of Hartree type equations and prove a quantitative blow up rate for their blow up solutions. This is an analogue of the result by Merle and Raphaël on 3d NLS.
This note studies the asymptotic behavior of global solutions to the fourth-order generalized Hartree equation $$i\dot u+Δ^2 u\pm(I_α*|u|^p)|u|^{p-2}u=0.$$ Indeed, for both attractive and repulsive sign, the scattering is obtained in the mass super-critical and energy sub-critical regimes, with radial setting.
The energy-critical inhomogeneous generalized Hartree equation in 3D
本文研究3D能量临界非齐次广义Hartree方程，利用非线性衰减性与Kenig-Merle路线图，在非径向初值下建立基态阈值以下全局适定性与散射，并引入新技术克服势与卷积型非局部非线性项的挑战。
Scattering for the non-radial inhomogeneous Hartree equation with a potential
本文研究带势且含 |x|^{-b} 权的聚焦非齐次 Hartree 方程，借助受扰算子、Tao 散射准则和局部 Morawetz 估计，证明非径向初值在临界间情形下的散射及小数据全局适定性。
Energy-critical inhomogeneous generalized Hartree equation with inverse square potential
本文研究带逆平方势的能量临界非齐次广义Hartree方程的柯西问题，建立适定性理论与基态阈值下解的二分性，解决非局部源项等三个技术难题，填补能量临界情形研究空白。
On well-posedness for inhomogeneous Hartree equations in the critical case
本文利用 Sobolev-Lorentz 空间建立了非齐次 Hartree 方程在临界指标且 $1/2\le s\le1$（含能量临界）情形下的适定性理论，解决了此前未决问题。
By $I$-method, the interaction Morawetz estimate, long time Strichartz estimate and local smoothing effect of Schrödinger operator, we show global well-posedness and scattering for the defocusing Hartree equation $$\left\{ \begin{array}{ll} iu_t + Δu &=F(u), \quad (t,x) \in \mathbb{R} \times \mathbb{R}^4 u(0) \\ &=u_0(x)\in H^s(\mathbb{R}^4), \end{array} \right. $$ where $F(u)= (V* |u|^2) u$, and $V(x)=|x|^{-γ}$, $3< γ<4$, with radial data in $H^{s}(\mathbb{R}^4)$ for $s>s_c:=γ/2-1$. It is a sharp global result except of the critical case $s=s_c$, which is a very difficult open problem.
We prove scattering below the ground state threshold for an energy-critical inhomogeneous nonlinear Schrödinger equation in three space dimensions. In particular, we extend results of Cho, Hong, and Lee from the radial to the non-radial setting.
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…
We extend the result of Farah and Guzmán on scattering for the $3d$ cubic inhomogeneous NLS to the non-radial setting. The key new ingredient is a construction of scattering solutions corresponding to initial data living far from the origin.
In this paper, we study the long-time behavior of global solutions to the Schrödinger-Choquard equation $$i\partial_tu+Δu=-(I_α\ast|\cdot|^b|u|^{p})|\cdot|^b|u|^{p-2}u.$$ Inspired by Murphy, who gave a simple proof of scattering for the non-radial inhomogeneous NLS, we prove scattering theory below the ground state for the intercritical case in energy space without radial assumption.
In this paper, we consider the scattering theory of the radial solution to focusing energy-subcritical Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$ using the method from \cite{Dodson2016}. The main difficulties are the equation is \emph{not} space-translation invariant and the nonlinearity is non-local. Using the radial Sobolev embedding and a virial-Morawetz type estimate we can exclude the concentration of mass near the origin. Besides, we can overcome the weak dispersive estimate when $a<0$, using the dispersive estimate established by \cite{zheng…
Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
本文研究广义能量临界Hartree方程径向阈值解的长时间动力学，通过证明非局部结构正泡解的Ḣ^1非退化性，结合谱理论、调制分析与集中紧性刚性论证，完成了所有径向阈值解的分类。
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 $\mathbb{R}\times \mathbb{R}^d$, $d\geq 5$.
This note studies the asymptotic behavior of global solutions to the fourth-order Schrödinger equation $$i\dot u+Δ^2 u+F(x,u)=0 .$$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to Dodson-Murphy \cite{dm}.
We consider the $L^2$ critical inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$ i \partial_t u +Δu +|x|^{-b} |u|^{\frac{4-2b}{N}}u = 0, $$ where $N\geq 1$ and $0<b<2$. We prove that if $u_0\in H^1(\mathbb{R}^N)$ satisfies $E[u_0]<0$, then the corresponding solution blows-up in finite time. This is in sharp contrast to the classical $L^2$ critical NLS equation where this type of result is only known in the radial case for $N\geq 2$.
We study long time dynamics of non-radial solutions to the focusing inhomogeneous nonlinear Schrödinger equation. By using the concentration/compactness and rigidity method, we establish a scattering criterion for non-radial solutions to the equation. We also prove a non-radial blow-up criterion for the equation whose proof makes use of localized virial estimates. As a byproduct of these criteria, we study long time dynamics of non-radial solutions to the equation with data lying below, at, and above the ground state threshold. In addition, we provide a new argument showing the existence of fi…
We prove scattering below the mass-energy threshold for the focusing inhomogeneous nonlinear Schrödinger equation \begin{equation} iu_t + Δu + |x|^{-b}|u|^{p-1}u=0, \end{equation} when $b \geq 0$ and $N > 2$ in the intercritical case $0 < s_c <1$. This work generalizes the results of Farah and Guzmán [9], allowing a broader range of values for the parameters $p$ and $b$. We use a modified version of Dodson-Murphy's approach [6], allowing us to deal with the inhomogeneity. The proof is also valid for the classical nonlinear Schrödinger equation ($b = 0$), extending the work in [6] for radial so…
In the present article we study the radial symmetry of minimizers of the energy functional, corresponding to the repulsive Hartree equation in external Coulomb potential. To overcome the difficulties, resulting from the "bad" sign of the nonlocal term, we modify the reflection method and then, by using Pohozaev integral identities we get the symmetry result.
Isolated Singularities for Fractional Hartree Equations
研究带Riesz相互作用的分数阶Hartree方程正解的孤立奇点，通过Riesz分解和Kelvin球面移动证明径向对称性和单调性，并构造显式齐次奇点解。
We investigate the focusing $\dot H^{1/2}$-critical nonlinear Schrödinger equation (NLS) of Hartree type $i\partial_t u + Δu = -(|\cdot|^{-3} \ast |u|^2)u$ with $\dot H^{1/2}$ radial data in dimension $d = 5$. It is proved that if the maximal life-span solution obeys $\sup_{t}\big\||\nabla|^{1/2}u\big\|_2 < \frac{\sqrt{6}}{3} \big\||\nabla|^{1/2}Q\big\|_2$, where $Q$ is the positive radial solution to the elliptic equation(\ref{e14}). Then the solution is global and scatters.
We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Hartree equation $iu_t+Δu=\pm(|x|^{-2}*|u|^2)u$ for large spherically symmetric $L^2_x(\Bbb{R}^d)$ initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state.
We consider the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b}|u|^αu = 0\quad\text{on}\quad\mathbb{R}\times\mathbb{R}^N, \] with $N\geq 2$, $0<b<\min\{\tfrac{N}{2},2\}$, and $\tfrac{4-2b}{N}<α<\tfrac{4-2b}{N-2}$. These constraints make the equation mass-supercritical and energy-subcritical. We extend the results of Farah-Guzmán and Miao-Murphy-Zheng and prove scattering below the ground state with general initial data.
In this paper we consider the inhomogeneous nonlinear Schrödinger (INLS) equation \begin{align}\label{inls} i \partial_t u +Δu +|x|^{-b} |u|^{2σ}u = 0, \,\,\, x \in \mathbb{R}^N \end{align} with $N\geq 3$. We focus on the intercritical case, where the scaling invariant Sobolev index $s_c=\frac{N}{2}-\frac{2-b}{2σ}$ satisfies $0<s_c<1$. In a previous work, for radial initial data in $\dot H^{s_c}\cap \dot H^1$, we prove the existence of blow-up solutions and also a lower bound for the blow-up rate. Here we extend these results to the non-radial case. We also prove an upper bound for the blow-up…
Consider the focusing inhomogeneous nonlinear Schrödinger equation in $H^1(\mathbb{R}^N)$, $$iu_t + Δu + |x|^{-b}|u|^{p-1}u=0,$$ when $b > 0$ and $N \geq 3$ in the intercritical case $0 < s_c <1$. In previous works, the second author, as well as Farah, Guzmán and Murphy, applied the concentration-compactness approach to prove scattering below the mass-energy threshold for radial and non-radial data. Recently, the first author adapted the Dodson-Murphy approach for radial data, followed by Murphy, who proved scattering for non-radial solutions in the 3d cubic case, for $b<1/2$. This work genera…
Blow-up results for Inhomogeneous fourth-order nonlinear Schrödinger Equation
本文研究非齐次双调和薛定谔方程解的H²范数爆破现象，通过凸性方法和virial恒等式，分别分析负能量及低于基态能量且动能超基态的正能量下径向与非径向解的爆破情况。
In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schrödinger equation in the radial case. Then we apply them to the well-posedness theory for the equation $i\partial_{t}u+|\nabla|^αu=V(x,t)u$, $1<α<2$, with radial $\dot{H}^γ$ initial data below $L^2$ and radial potentials $V\in L_t^rL_x^w$ under the scaling-critical range $α/r+n/w=α$.
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 first take advantage of the term $\displaystyle - \int_{I}\int_{|x|\leq A|I|^{1/2}}|u|^{2}Δ\Big(\frac{1}{|x|}\Big)dxdt$ in the localized Morawetz identity to rule out the possibility of energy concentration, instead of the classical Morawetz estimate dependent of the nonlinearity.
The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schrödinger equation $$ i u_t +Δu+|x|^{-b}|u|^2 u = 0, $$ where $0<b<1/2$. Let $Q$ be the ground state solution of $-Q+ΔQ+ |x|^{-b}|Q|^{2}Q=0$ and $s_c=(1+b)/2$. We show that if the radial initial data $u_0$ belongs to $H^1(\mathbb{R}^3)$ and satisfies $E(u_0)^{s_c}M(u_0)^{1-s_c}<E(Q)^{s_c}M(Q)^{1-s_c}$ and $\| \nabla u_0 \|_{L^2}^{s_c} \|u_0\|_{L^2}^{1-s_c}<\|\nabla Q \|_{L^2}^{s_c} \|Q\|_{L^2}^{1-s_c}$, then the corresponding solution is global and scatters in $H^1(\mathbb{R}^3)$. Our proof is based in the ideas int…
Scattering for the mass-critical fractional Schrödinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is established. For this, we prove a bilinear estimate for free solutions and extend it to perturbations of bounded quadratic variation. This result is shown to be sharp by proving the unboundedness of a third order derivative of the flow map in the super-critical range.
Global dynamics above the ground state for the energy-critical Hartree equation with radial data
本文研究径向数据下能量临界 Hartree 方程在能量略高于基态时的全局动力学，通过分析基态附近的双曲动力学行为和单次通过引理，将 Nakanishi 和 Schlag 在次临界非线性 Klein-Gordon 方程中的方法推广到能量临界情形，从而对径向解进行完整分类。
Global Well-posedness and Scattering for the Focusing Energy-critical Inhomogeneous Nonlinear Schrödinger Equation with Non-radial Data
本文利用集中紧性论证和Lorentz空间发展稳定性理论，证明了聚焦能量临界非齐次非线性薛定谔方程在非径向数据下的整体适定性与散射，统一了聚焦与散焦情况并放宽了维度参数限制。
We prove existence of positive ground state solutions to the pseudo-relativistic Schrödinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-Δ+m^2} u +Vu = \left( W * |u|^θ \right)|u|^{θ-2} u \quad\text{in $\mathbb{R}^N$}\\ u \in H^{1/2}(\mathbb{R}^N) \end{array} \right. \end{equation*} where $N \geq 3$, $m >0$, $V$ is a bounded external scalar potential and $W$ is a convolution potential, radially symmetric, satisfying suitable assumptions. We also furnish some asymptotic decay estimates of the found solutions.
Global Dynamics of the Non-Radial Energy-Critical Inhomogeneous Biharmonic NLS
本文研究能量临界非齐次双调和NLS在非径向情形下的全局适定性与散射，通过发展基于Kenig-Merle方法的集中紧性与刚性框架，在次临界动能假设下克服了空间非齐次与四阶色散算子带来的分析困难。
We study finite-time blow-up for pseudo-relativistic Hartree- and Hartree-Fock equations, which are model equations for the dynamical evolution of white dwarfs. In particular, we prove that radially symmetric initial configurations with negative energy lead to finite-time blow-up of solutions. Furthermore, we derive a mass concentration estimate for radial blow-up solutions. Both results are mathematically rigorous and are in accordance with Chandrasekhar's physical theory of white dwarfs, stating that stellar configurations beyond a certain limiting mass lead to ``gravitational collapse'' of …
Blow-up for the 3D intercritical inhomogeneous NLS with inverse-square potential
本文针对质量超临界、能量次临界区域中带反平方势的聚焦非齐次三维非线性薛定谔方程，建立了临界空间中的局部适定性，并证明了径向及非径向解的标度不变Lebesgue范数爆破。
We adapt the argument of Dodson-Murphy to give a simple proof of scattering below the ground state for the intercritical inhomogeneous nonlinear Schrödinger equation. The decaying factor in the nonlinearity obviates the need for a radial assumption.
We consider the final state problem for the inhomogeneous nonlinear Schrödinger equation with a critical long-range nonlinearity. Given a prescribed asymptotic profile, which has a logarithmic phase correction compared with the free evolution, we construct a unique global solution which converges to the profile. As a consequence, the existence of modified wave operators for localized small scattering data is obtained. We also study the same problem for the case with the critical inverse-square potential under the radial symmetry. In particular, we construct the modified wave operators for the …
Determination of radial nonlocal nonlinearities from the scattering map
本文证明对于一类具有径向Hartree型非线性的非线性薛定谔方程，小数据散射图唯一确定非线性项，卷积核仅需在无穷远处有轻微衰减且允许原点处局部可积奇异性。
We consider the inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$i \partial_t u +Δu +|x|^{-b} |u|^{2σ}u = 0,$$ where $N\geq 3$, $0<b<\min\left\{\frac{N}{2},2\right\}$ and $\frac{2-b}{N}<σ<\frac{2-b}{N-2}$. The scaling invariant Sobolev space is $\dot{H}^{s_c}$ with $s_c=\frac{N}{2}-\frac{2-b}{2σ}$. The restriction on $σ$ implies $0<s_c<1$ and the equation is called intercritical (i.e. mass-supercritical and energy-subcritical). Let $u_0\in \dot H^{s_c}\cap \dot H^1$ be a radial initial data and $u(t)$ the corresponding solution to the INLS equation. We first show that if…
Non-radial Blow-up for a mass-critical fourth-order inhomogeneous nonlinear Schrödinger equation
本文研究了质量临界四阶非齐次非线性薛定谔方程的非径向解在负能量下的有限或无限时间破裂现象。
We consider a class of $L^2$-supercritical inhomogeneous nonlinear Schrödinger equations with potential in three dimensions \[ i\partial_t u + Δu - V u = \pm |x|^{-b} |u|^αu, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^3, \] where $0<b<1$ and $α>\frac{4-2b}{3}$. In the focusing case, by adapting an argument of Dodson-Murphy, we first study the energy scattering below the ground state for the equation with radially symmetric initial data. We then establish blow-up criteria for the equation whose proof is based on an argument of Du-Wu-Zhang. In the defocusing case, we also prove the energy scat…
Delaunay solutions to the fractional Hartree equation with critical growth
本文研究带原点不可移除孤立奇点的临界分数阶Hartree方程，结合Caffarelli–Silvestre延拓与移动球方法证明其正奇异解径向对称，并通过最小化Rayleigh型商构造出所有足够大周期的Delaunay型周期奇异解。
We consider a class of $L^2$-supercritical inhomogeneous nonlinear Schrödinger equations in two dimensions \[ i\partial_t u + Δu = \pm |x|^{-b} |u|^αu, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2, \] where $0<b<1$ and $α>2-b$. By adapting a new approach of Arora-Dodson-Murphy \cite{ADM}, we show the energy scattering for the equation with radially symmetric initial data. In the focusing case, our result extends the one of Farah-Guzmán \cite{FG-high} to the whole range of $b$ where the local well-posedness is available. In the defocusing case, our result extends the one in \cite{Dinh-scat} w…
Scattering Criteria for the Three-Dimensional Focusing Energy-Critical Generalized Hartree Equation
针对三维聚焦能量临界广义Hartree方程，在基态阈值下建立两个非径向散射准则，通过结合局域Hartree位力恒等式与临界Hardy-Littlewood-Sobolev估计完成证明。
We study the Cauchy problem for fractional Schrödinger equation with cubic convolution nonlinearity ($i\partial_t u - (-Δ)^{\fracα{2}}u\pm (K\ast |u|^2) u =0$) with Cauchy data in the modulation spaces $M^{p,q}(\mathbb R^{d}).$ For $K(x)= |x|^{-γ}$ $ (0< γ< \text{min} \{α, d/2\})$, we establish global well-posedness results in $M^{p,q}(\mathbb R^{d}) (1\leq p \leq 2, 1\leq q < 2d/ (d+γ))$ when $α=2, d\geq 1$, and with radial Cauchy data when $d\geq 2, \frac{2d}{2d-1}< α< 2. $ Similar results are proven in Fourier algebra $\mathcal{F}L^1(\mathbb R^d) \cap L^2(\mathbb R^d).$
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 Duyckaerts-Roudenko [11] for the $3$d focusing cubic nonlinear Schrödinger equation (NLS). In particular, apart from the ground state solution $Q$, which is global but non-scattering, there exist special solutions $Q^+$ and $Q^-$, which in one time direction approach $Q$ exponentially, and in the other time direction $Q^+$ blows up in finite time and $Q^-$ …
We consider the focusing inhomogeneous mass critical half-wave equation in one dimension. Under the mild conditions of the inhomogeneous factor, we show that the existence of the radial blowup solutions with ground state mass $\|u_0\|_2=\|Q\|_2$, where $Q$ is the unique positive ground state solution of equation $DQ+Q=Q^3$ and obtain the blowup rate $\|D^{\frac{1}{2}}u(t)\|_{L^2}\sim\frac{1}{|t|}$ as $t\nearrow 0^-$.
We consider the focussing energy-critical inhomogeneous nonlinear Schrödinger equation: $$ iu_t + Δu + g|u|^2u = 0, u(0)= φ\in \dot{H}^1,\;\; 0 \le g_i \le |x|g \le g_s.$$ On the road map of Kenig-Merle \cite{km} we show the global well-posedness and scattering of radial solutions under energy condition $$E_g(φ) < E_g(Q),\;\;\mbox{and}\;\; g_s\|φ\|_{\dot{H}^1}^2 < \|Q\|_{\dot{H}^1}^2,$$ where $Q$ is the solution of $ΔQ + |x|^{-1}Q^3 = 0$, together with scaling condition $|g(x)| + |x||\nabla g(x)| \lesssim |x|^{-1}$, variational condition $g_s(2-g_i) \le 1$, and rigidity condition $-g(x) \le x\…
Solutions to critical equations with a superposition of nonlocal Hartree-type nonlinearities
本文研究由Riesz势加权测度驱动的非局部椭圆方程，通过引入叠加Coulomb空间和变分方法，结合基于尺度的临界点理论，证明了径向解的存在性和多重性，并推广了Brezis-Nirenberg型问题。
We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation $i \partial_t u + Δu = - |x|^b |u|^{p-1} u$ in the radial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$. We show the global existence and energy scattering in the inter-critical regime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. …
Global well-posedness and scattering for mass-critical Hartree equation
本文在标度临界正则性下证明了质量临界 Hartree 方程的散射猜想，去除了径向限制，涵盖散焦与聚焦（质量小于基态）情形。
Classification of minimal-mass blowup solutions to the mass-critical Hartree equation at the $L^2$-critical level
该文对质量临界Hartree方程在L²临界水平下的最小质量爆破解进行分类，证明全局最小质量非散射解为孤波、有限时间最小质量爆破解为孤波的伪共形像，推广了相关结果并消除了径向假设。
Fractional nonlinear Schrödinger and Hartree equations in modulation spaces
本文在调制空间中针对质量次临界分数阶非线性薛定谔方程，对幂型与哈特里型非线性项建立了径向初值条件下的全局适定性。
In this paper, we consider a class of the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b} |u|^αu = 0, \quad u(0)=u_0 \in H^1(\mathbb{R}^d), \] with $0<b<\min\{2,d\}$ and $α_\star\leq α<α^\star$ where $α_\star =\frac{4-2b}{d}$ and $α^\star=\frac{4-2b}{d-2}$ if $d\geq 3$ and $α^\star = \infty$ if $d=1,2$. In the mass-critical case $α=α_\star$, we prove that if $u_0$ has negative energy and satisfies either $xu_0 \in L^2$ with $d\geq 1$ or $u_0$ is radial with $d\geq 2$, then the corresponding solution blows up in finite time. Moreover, when $d=1$, we prove …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。