奇异半线性抛物方程初值问题解的存在性与不存在性,Blow-up问题及解的无限增长性

该文研究一类奇异半线性抛物方程初值问题的非负局部解存在与不存在的条件，解的BloW-Up问题及当t时解的无限增长性．     

解非线性互补问题的光滑牛顿方法

由于退化解会导致再生方程的奇异性,非线性互补问题的求解通常采用基于半光滑技术的广义牛顿法.基于2-正则性的概念,提出了一类利用光滑互补函数求解互补问题的光滑牛顿算法.算法采用积极集技术,能在解的附近估计出退化指标,并把原问题降阶为一个非奇异方程组,从而保证了迭代效率.算法具有整体收敛性和局部超线性收敛性,数值实验显示算法是有效的.     

无界域上半线性奇异椭圆方程的共振与非共振问题

本文考虑了无界域上一类半线性奇异椭圆方程的加权特征值问题,然后讨论了在非共振与共振情形下这类方程非平凡解的存在性．     

奇异非对称两点边值问题的有限元解的整体高精度

有限元解的渐近展式是外推法的理论基础,同时也可用来研究有限元的超收敛、校正法及后验误差估计等。对于奇异系数问题,文[6]首先对线性情形f(x,u)=c(x)u+g(x),证明了均匀网格上的线性元解具有如下渐近展式:     

一类奇异半线性椭圆型方程的Dirichlet问题

得到了一类奇异半线性椭圆型方程 Ｄｉｒｉｃｈｌｅｔ问题解的存在性．     

一般形式的一阶椭圆组的非线性Riemann问题

讨论一般形式的一阶线性和拟线性椭圆型方程的非线性Riemann问题,通过把这些问题转化为奇异积分方程,利用压缩原理和广义压缩原理来证明在某些假设条件下所讨论问题的解的存在性．     

奇异半线性及超线性一维p-Laplacian方程边值问题的正解

本文讨论了奇异边值问题其中Φ（s）=|s|p-2s,p>1.函数g在u=0具有奇性,允许在u=∞处半线性或超线性,且可变号.解的存在性由上下解方法可得.     

线性奇异系统的解耦问题

本文讨论在状态反馈下的线性奇异系统的解耦问题，利用奇异系统的受限等价变换，给出了输出，输入维数相同的方线性奇异系统解耦问题有解的充分必要条件。     

理想弹塑性Ⅲ型扩展裂纹的全新和精确分析

本文采用线场分析方法对理想弹塑性Ⅲ型准静态扩展裂纹进行了分析.本文的意义在于突破了小范围屈服理论的限制.通过求得裂纹线附近塑性区应力和位移率的通解,并将此通解(而不是过去一直采用的特解)与弹性场的精确解(而不是线弹性奇异K场)在裂纹线附近的弹塑性边界上匹配,本文得出了裂纹线附近塑性区的应力变形场、塑性区的长度及弹塑性边界的单位法向量的全新和精确解答.本文的分析放弃了小范屈服理论的所有近似假定并且不再附加任何其它的近似假定,本文的结果在裂纹线附近是足够精确的.本文的结果表明:对理想弹塑性Ⅲ型准静态扩展裂纹,不存在“定常扩展状态”,且裂纹线附近塑性应变不存在奇异性.本文还对裂纹稳定扩展过程讨论了两种重要情形.     

一类Kirchhoff型p(t)-Laplace系统的无穷多周期解北大核心CSCD

利用变指数Sobolev空间理论和临界点理论,研究Kirchhoff型p(t)-Laplace系统的周期解.当非线性项在零点附近次线性或在无穷远处超线性增长时,得到了此类系统无穷多个周期解的存在性.     

偏微分方程的调和分析方法简介

本文致力于阐述调和分析与现代偏微分方程研究的关系,特别是奇异积分算子、拟微分算子、Fourier限制性估计、Fourier频率分解方法在椭圆边值问题、非线性发展方程研究中的重要作用.对于偏微分方程研究的各种方法进行了比较与分析,指出了偏微分方程的调和分析方法的优点与局限性.与此同时,还给出了偏微分方程的调和分析方法这一领域的最新研究进展.     

Zakharov–Shabat Spectral Transform on the Half-Line

The Zakharov–Shabat inverse spectral problem is constructed for a potential with support on the half-line and with a boundary value at the origin. This prescribed value is shown to produce a Jost solution with an essential singularity at large values of the spectral parameter; this requires particular attention when solving the related Hilbert boundary value problem. The method is then used to illustrate the sine-Gordon equation (in the light cone) and is discussed using a singular limit of the stimulated Raman scattering equations.     

Simple layer potential method for domains having external corners

This note investigates the simple layer potential method for domains having external corners. The singular behaviour of simple layer density at the corners is studied for the Neumann problem of Helmholtz's equation. A numerical technique of solving the integral equation for this problem is proposed. This technique takes the singularity of the solution into consideration. Some numerical examples are given to show the applicability of the present method.     

Solutions with moving singularities for a semilinear parabolic equation

We consider the Cauchy problem for a semilinear heat equation with power nonlinearity. It is known that the equation has a singular steady state in some parameter range. Our concern is a solution with a moving singularity that is obtained by perturbing the singular steady state. By formal expansion, it turns out that the remainder term must satisfy a certain parabolic equation with inverse-square potential. From the well-posedness of this equation, we see that there appears a critical exponent. Paying attention to this exponent, for a prescribed motion of the singular point and suitable initial data, we establish the time-local existence, uniqueness and comparison principle for such singular solutions. We also consider solutions with multiple singularities.     

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.     

On singular solutions of the initial boundary value problem for the Stokes system in a power cusp domain

The initial boundary value problem for the non-steady Stokes system is considered in bounded domains with the boundary having a peak-type singularity (power cusp singularity). The case of the boundary value with a nonzero time-dependent flow rate is studied. The formal asymptotic expansion of the solution near the singular point is constructed. This expansion contains both the outer asymptotic expansion and the boundary-layer-in-time corrector with the ‘fast time’ variable depending on the distance to the cusp point. The solution of the problem is constructed as the sum of the asymptotic expansion and the term with finite energy.     

Jacobi Fields Along Harmonic 2-Spheres in CP2 are Integrable

The paper shows that any Jacobi field along a harmonic map fromthe 2-sphere to the complex projective plane is integrable (thatis, is tangent to a smooth variation through harmonic maps).This provides one of the few known answers to the problem ofintegrability, which was raised in different contexts of geometryand analysis. It implies that the Jacobi fields form the tangentbundle to each component of the manifold of harmonic maps fromS² to CP² thus giving the nullity of any such harmonic map;it also has a bearing on the behaviour of weakly harmonic E-minimizingmaps from a 3-manifold to CP² near a singularity and the structureof the singular set of such maps from any manifold to CP².     

A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces

We construct a harmonic diffeomorphism from the Poincaré ballH ⁿ⁼¹ to itself, whose boundary value is the identity on the sphereS ⁿ, and which is singular at a boundary point, as follows: The harmonic map equations between the corresponding upper-half-space models reduce to a nonlinear o.d.e. in the transverse direction, for which we prove the existence of a solution on the whole R₊ that grows exponentially near infinity and has an expansion near zero. A conjugation by the inversion brings the singularity at the origin, and a conjugation by the Cayley transform and an isometry of the ball moves the singularity at any point on the sphere.     

Analysis of the singular function boundary integral method for a biharmonic problem with one boundary singularity

In this article, we analyze the singular function boundary integral method (SFBIM) for a two‐dimensional biharmonic problem with one boundary singularity, as a model for the Newtonian stick‐slip flow problem. In the SFBIM, the leading terms of the local asymptotic solution expansion near the singular point are used to approximate the solution, and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. By means of Green's theorem, the resulting discretized equations are posed and solved on the boundary of the domain, away from the point where the singularity arises. We analyze the convergence of the method and prove that the coefficients in the local asymptotic expansion, also referred to as stress intensity factors, are approximated at an exponential rate as the number of the employed expansion terms is increased. Our theoretical results are illustrated through a numerical experiment. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

The problem of an earth dam

The behaviour of the solution of the problem of a rectangular earth dam in the neighbourhood of a singular point at the intersection of the free surface and the seepage area is investigated. A similar result is obtained for an earth dam with a slanting lower incline.