On a wave equation with supercritical interior and boundary sources and damping terms

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well‐posed in the finite‐energy space H¹(Ω) × L₂(?Ω) with boundary data in L₂ due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non‐dissipative and are not locally Lipschitz operators from H¹(Ω) into L₂(Ω), or L₂(?Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.     

Mathematica软件在微分流形中的应用

本文利用Mathematica软件描绘向量场和流,通过具体的例子,探讨该软件在微分流形教学科研中的应用.     

改进的PNC方法及其在非线性偏微分方程多解问题中的应用

2017年,李昭祥等提出了一种偏牛顿-校正法（Partial Newton-Correction Method,简记为PNC方法）,并利用它成功地计算出了三类非线性偏微分方程的多重不稳定解.本文在PNC方法的基础上,提出并发展了一种改进的PNC方法.首先,利用Nehari流形$\mathcal{N}$与零平凡解的可分离性,建立并证明了$\mathcal{N}$的某特殊子流形$\mathcal{M}$上的全局分离定理及其推广（即局部分离定理）.全局分离定理只跟非线性偏微分算子或相应的非线性泛函本身有关,而与具体的计算方法无关.对一些典型的非线性偏微分方程多解问题（比如,Henon方程问题）,该全局分离定理的分离条件,经验证是成立的.另一个方面,通过修改或补充原辅助变换的定义,去掉了原辅助变换的奇异性;接着建立并证明了某些非线性偏微分方程问题的新未知解与该非线性偏微分算子零核空间的密切关系;在证明中,去掉了在原奇异变换下所需的标准收敛（standard convergence）假设.最后,计算实例与数值结果验证了改进的PNC方法的可行性和有效性;同时表明子流形$\mathcal{M}$与已知解的可分离性是PNC方法和本文新方法能成功找到多解的关键.     

Canonical Connections and Algebraic Ricci Solitons of Three-dimensional Lorentzian Lie Groups

In this paper, the author computes canonical connections and KobayashiNomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. He defines algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. He classifies algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.     

含有陡峭势阱和凹凸非线性项的Kirchhoff型问题的多重正解*

在这篇文章中, 作者研究涉及凹凸非线性项的Kirchhoff型问题-(a + b ∫R3|▽u|²dx) Δu + λV (x)u = μf(x)|u|^q?2u + |u|^p?2u, x ∈ R³,u ∈ H¹(R³),其中a,b > 0 是常数, λ, μ > 0 是参数, 1 < q < 2, 4 < p < 6 且 V 是一个非负连续位势. 在f(x) 和 V 的合适条件下,此问题正解的存在性和集中性能够通过Nehari 流形和Ekeland 变分原理得到.     

一类非线性不确定系统的结构性质

考察一类带有匹配不确定性的非线性系统，采用微分拓扑方法，将标称系统关于局部能控分解的结果和（f，g）-不变分布的Quaker引理推广到非线性不确定性系统．     

负曲率完备流形上方程△u=f的有界整体解

借助于Hessian比较定理和上下解方法，证明了对于相当多的一类非负函数f，负曲率完备黎曼流形上的方程△u=f都存在有界的整体解，从而刻画了这些黎曼流形的双曲性程度。     

The geography of irreducible 4-manifolds

In this paper we investigate the existence and the uniqueness problems for simply connected irreducible -manifolds. By taking fiber sums along an embedded surface of square and by a rational blow-down procedure, we construct many new irreducible -manifolds which have infinitely many distinct smooth structures. Furthermore, we prove that all but at most finitely many lattice points lying in the non-positive signature region with are covered by these irreducible -manifolds.

     

Quaternionic Killing Spinors

In [17] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kähler manifolds. In the present article we study the limiting case, i.e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and show that the only symmetric quaternionic Kähler manifolds with smallest possible eigenvalue are the quaternionic projective spaces.     

Asymptotics of the parabolic Green function for an elliptic operator on a manifold with conical points

We construct the asymptotics ast→0 of the trace of the operator exp(−tP) for an elliptic operatorP on a manifold with conical points. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 28–36, January, 1998. This research was supported by the Belarus Foundation for Basic Research under grant No. 96-01 00790.