Walter H. Aschbacher, Marco Squassina
We prove the phase segregation phenomenon to occur in the ground state solutions of an interacting system of two self-coupled repulsive Hartree equations for large nonlinear and nonlocal interactions. A self-consistent numerical investigation visualizes the approach to this segregated regime.
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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。
Uniqueness of Ground State Solutions for a Defocusing Hartree Equation via Inverse Optimal Problems
本文通过逆最优问题方法，证明了非聚焦哈特里方程地面态解的存在性和唯一性，并探讨了主解的连续依赖性和双变分公式。
Fully discrete Galerkin schemes for the nonlinear and nonlocal Hartree equation
研究连续、半离散和全离散框架下时间依赖Hartree方程的存在唯一性、正则性和逼近性质，为物理应用中Hartree动力学的非线性非局部特征的可控数值计算奠定基础。
In this paper, we are concerned with static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities. We derive the explicit forms for positive solution $u$ in the critical case and non-existence of nontrivial nonnegative solutions in the subcritical cases (see Theorem \ref{Thm0} and \ref{Thm1}). The arguments used in our proof is a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians in \cite{CLZ}. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., \emph{Narrow region princip…
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 study the existence of ground state solutions for a class of non-linear pseudo-relativistic Schrödinger equations with critical two-body interactions. Such equations are characterized by a nonlocal pseudo-differential operator closely related to the square-root of the Laplacian. We investigate such a problem using variational methods after transforming the problem to an elliptic equation with a nonlinear Neumann boundary conditions.
Ground state solutions of mixed local-nonlolcal equations with Hartree type nonlinearities
本文研究了混合局部-非局部方程的基态解存在性，通过正则性性质、Pohožaev恒等式和极化方法证明了其存在性。
In this paper, we consider a nonlocal advection model for two populations on a bounded domain. The first part of the paper is devoted to the existence and uniqueness of solutions and the associated semi-flow properties. Here we use the notion of solution integrated along the characteristics. Next, by proving segregation property, we construct an energy functional to investigate the asymptotic behavior of the solution. In order to get some compactness of the positive orbit, we use the narrow convergence in the space of Young measures. By using this idea, we get a description of the asymptotic b…
Existence and Regularity in the Small-Mass Regime for a Hartree--Ohta-Kawasaki Shape Optimization Problem
本文研究了混合能函数的小质量范围内形状优化问题，证明了体积约束下的极小值解存在且为球体的C^{2,α}扰动，结合手术技术、Γ收敛、椭圆PDE理论和一相自由边界正则性。
Construction of two-bubble solutions for the energy-critical Hartree equation
本文构造了能量临界Hartree方程的纯双气泡解，通过调制分析、Bootstrap论证和拓扑论证，解决了非局部相互作用带来的分析难题。
Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent
本文研究了非局部方程中稍亚临界指数单破裂解的定性性质，通过局部Pohozaev恒等式和破裂分析，提供了特征值估计并推导了单气泡解的Morse指数。
We consider the semiclassical limit of the Hartree equation with a data causing a focusing at a point. We study the asymptotic behavior of phase function associated with the WKB approximation near the caustic when a nonlinearity is supercritical. In this case, it is known that a phase shift occurs in a neighborhood of focusing time in the case of focusing cubic nonlinear Schrödinger equation. Thanks to the smoothness of the nonlocal nonlinearities, we justify the WKB-type approximation of the solution for a data which is larger than in the previous results and is not necessarily well-prepared.…
Ground state solutions for Hartree type equations driven by superposition operators and Pohozaev Identity
研究由非局部算子驱动的Hartree型方程，在特定假设下证明山路解存在及能量水平与Pohozaev流形最小值一致，建立非负解有界性，还得到一般Pohozaev恒等式，核心方法包括使用Sobolev嵌入、精细处理积分及Kato型不等式。
This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. The local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors; the nonlocal model studied here consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interv…
We investigate the NLS equation with competing Hartree-type and power-type nonlinearities \begin{equation*} \begin{array}{ll} i\partial _{t}ψ+Δψ+γ(I_{α}\ast |ψ|^{p})|ψ|^{p-2}ψ+μ|ψ|^{q-2}ψ=0, & \text{ }\forall (t,x)\in \mathbb{R\times R}^{N},% \end{array}% \end{equation*}% where $γμ<0$. We establish conditions for the local well-posedness in the energy space. Under the double $L^{2}$-supercritical case, we prove the existence and multiplicity of standing waves with prescribed mass by developing a constraint method when $γ<0,μ>0$ and $γ>0,μ<0, $ respectively. Moreover, we prove weak orbital stab…
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 investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do no…
On fractional Schrödinger equations with Hartree type nonlinearities
本文研究一类含Hartree型非线性项的分数阶薛定谔方程，证明了基态解的存在性，并获得了通解的 regularity 与渐近衰减性质，推广了已有结果。
We study the coupled Hartree system $$ \left\{\begin{array}{ll} -Δu+ V_1(x)u =α_1\big(|x|^{-4}\ast u^{2}\big)u+β\big(|x|^{-4}\ast v^{2}\big)u &\mbox{in}\ \mathbb{R}^N,\\[1mm] -Δv+ V_2(x)v =α_2\big(|x|^{-4}\ast v^{2}\big)v +β\big(|x|^{-4}\ast u^{2}\big)v &\mbox{in}\ \mathbb{R}^N, \end{array}\right. $$ where $N\geq 5$, $β>\max\{α_1,α_2\}\geq\min\{α_1,α_2\}>0$, and $V_1,\,V_2\in L^{N/2}(\mathbb{R}^N)\cap L_{\text{loc}}^{\infty}(\mathbb{R}^N)$ are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ movin…
Long-time behaviour and bifurcation analysis of a two-species aggregation-diffusion system on the torus
本文研究了环面上非局部聚集-扩散方程的稳态存在性与稳定性，利用 Crandall & Rabinowitz 分岔理论对两物种系统进行严格分析，分类解分支并推导 Fréchet 导数，揭示了纯吸引作用下细胞分选起始时的稳定分离模式。
Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations (PDEs), these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection-diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual av…
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