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Sharpness on the Lower Bound of the Lifespan of Solutions to Nonlinear Wave Equations

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Chinese Annals of Mathematics,Series B 2011年第4期32卷 521-526页

作者： Yi ZHOU 1 Wei HAN 2 1 Nonlinear mathematical Modeling and Methods Laboratory shanghai Key Laboratory for Contem- porary Applied Mathematics school of mathematical sciences, fudan university, shanghai 200433, china. 2 school of mathematical sciences, fudan university, shanghai 200433, china.Department of Mathematics, North university of china. Taiyuan 030051, china. Nonlinear mathematical Modeling and Methods Laboratory shanghai Key Laboratory for Contemporary Applied Mathematics school of mathematical sciences Fudan University Shanghai China school of mathematical sciences Fudan University Shanghai China Department of Mathematics North University of China Taiyuan China

This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n = 2 and cubic nonlinearity （see the results of T. T. Li and Y. M. Chen in 1992）. For this purpose, the authors consider the following Cauchy problem： { where □=δt^2-∑i=1n δx^^2 is the wave operator, g（x）（≡/） 0 is a smooth non-negative function with compact support, and ε 〉 0 is a small parameter. It is shown that the solution blows up in a finite time, and the lifespan T（ε） of solutions has an upper bound T（ε） ≤ exp（Aε-2） with a positive constant A independent of ε, which belongs to the same kind of the lower bound of the lifespan.

关键词： Nonlinear wave equation Cauchy problem Lifespan

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions

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Chinese Annals of Mathematics,Series B 2007年第2期28卷 205-212页

作者： Yi ZHOU school of mathematical sciences Fudan University Shanghai 200433 China

In this paper, the author considers equations with critical exponent in n ≥4 space tions on the initial data, it is proved that there small the initial data are. the Cauchy problem for semilinear wave dimensions. Und... 详细信息

关键词： Semilinear wave equation Critical exponent Cauchy problem Blow up

THE DISTANCE BETWEEN DIFFERENT COMPONENTS OF THE UNIVERSAL TEICHMULLER SPACE

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Chinese Annals of Mathematics,Series B 2005年第4期26卷 537-542页

作者： WANG ZHE school of mathematical sciences Fudan University Shanghai 200433 China.

The model of the universal Teichmuller union of infinite disconnected components. between different components is 0, and the every other component is 2. space by the derivatives of logarithm is the In this paper, it i... 详细信息

关键词： Universal Teichmuller space Logarithmic derivative Quasiconformal extension

Global Classical Solutions to Partially Dissipative Quasilinear Hyperbolic Systems

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Chinese Annals of Mathematics,Series B 2011年第5期32卷 771-780页

作者： Yi ZHOU 11 Key Laboratory of Mathematics for Nonlinear sciences,Ministry of Education,china.shanghai Key Laboratory for Contemporary Applied Mathematics school of mathematical sciences,fudan university,shanghai 200433,china. Key Laboratory of Mathematics for Nonlinear sciences Ministry of Education China shanghai Key Laboratory for Contemporary Applied Mathematics school of mathematical sciences Fudan University Shanghai China

The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic *** the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solutions for small initial data (in *** ***.,Vol.169,2003,pp.89-117).The aim of this paper is to give a completely different proof of this result with slightly different assumptions.

关键词： Cauchy problem Global classical solution Partially dissipativequasilinear hyperbolic system

Global Exact Boundary Controllability for Cubic Semi-linear Wave Equations and Klein-Gordon Equations

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Chinese Annals of Mathematics,Series B 2010年第1期31卷 35-58页

作者： Yi ZHOU Wei XU Zhen LEI Key Laboratory of Mathematics for Nonlinear sciences School of Mathematical Sciences Fudan University Shanghai 200433 China school of mathematical sciences Fudan University Shanghai 200433 China

The authors prove the global exact boundary controllability for the cubic semilinear wave equation in three space dimensions,subject to Dirichlet,Neumann,or any other kind of boundary controls which result in the well-posedness of the corresponding initial-boundary value *** exponential decay of energy is first established for the cubic semi-linear wave equation with some boundary condition by the multiplier method,which reduces the global exact boundary controllability problem to a local *** proof is carried out in line with [2,15].Then a constructive method that has been developed in [13] is used to study the local *** when the region is star-complemented,it is obtained that the control function only need to be applied on a relatively open subset of the *** the cubic Klein-Gordon equation,similar results of the global exact boundary controllability are proved by such an idea.

关键词： Global exact boundary controllability Cubic semi-linear wave equations The exponential decay Star-shaped Star-complemented Cubic Klein- Gordon equations

Weierstrass Representation for Surfaces in the Three-Dimensional Heisenberg Group

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Chinese Annals of Mathematics,Series B 2010年第1期31卷 119-132页

作者： Qun CHEN Hongbing QIU school of Mathematics and Statistics Wuhan University Wuhan 430072 China Corresponding author. school of mathematical sciences Fudan University Shanghai 200433 China

The authors define the Gauss map of surfaces in the three-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean ***,a second order partial differential equation for the Gauss m... 详细信息

关键词： Heisenberg group Mean curvature Weierstrass representation

On Galois Extension of Hopf Algebras

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Chinese Annals of Mathematics,Series B 2008年第1期29卷 65-70页

作者： Guohua LIU Shenglin ZHU Department of Mathematics Southeast University Nanjing 210096 China Key Laboratory of Mathematics for Nonlinear sciences School of Mathematical Sciences Pudan University Shanghai 200433 China

Let H be a cosemisimple Hopf algebra over a field k, and π ： A→ H be a surjective cocentral bialgebra homomorphism of bialgebras. The authors prove that if A is Galois over its coinvariants B=LH Ker π and B is a sub-Hopf algebra of A, then A is itself a Hopf algebra. This generalizes a result of Cegarra [3] on group-graded algebras.

关键词： Hopf algebra Galois extension

Koszul Differential Graded Algebras and BGG Correspondence II

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Chinese Annals of Mathematics,Series B 2010年第1期31卷 133-144页

作者： Jiwei HE Quanshui WU Department of Mathematics Shaoxing College of Arts and Sciences Shaoxing 312000 Zhejiang China school of mathematical sciences Fudan University Shanghai 200433 China

The concept of Koszul differential graded (DG for short) algebra is introduced in [8].Let A be a Koszul DG *** the Ext-algebra of A is finite-dimensional,i.e.,the trivial module A k is a compact object in the derived category of DG A-modules,then it is shown in [8] that A has many nice ***,if the Ext-algebra is infinitedimensional,little is known about *** shown in [15] (see also Proposition 2.2),A k is not compact if H(A) is *** this paper,it is proved that the Koszul duality theorem also holds when H(A) is finite-dimensional by using Foxby duality.A DG version of the BGG correspondence is deduced from the Koszul duality theorem.

关键词： Koszul differential graded algebra Koszul duality BGG correspondence

Operations on 3-Dimensional Small Covers

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Chinese Annals of Mathematics,Series B 2010年第3期31卷 393-410页

作者： Shintaro KUROKI Department of mathematical Science KAIST 335 Gwahangno (373-1 Guseong-dong) Yuseong-gu Daejeon 305-701 Korea school of mathematical sciences Fudan University

The purpose of this paper is to study relations among equivariant operations on 3-dimensional small covers. The author gets three formulas for these operations. As an application, the Nishimura＇s theorem on the construction of oriented 3-dimensional small covers and the Lu-Yu＇s theorem on the construction of all 3-dimensional small covers are improved. Moreover, for a construction of 3-dimensional 2-torus manifolds, it is shown that all operations can be obtained by using the equivariant surgeries.

关键词： Equivariant surgery Finite group action Small cover 3-dimensional manifold 3-dimensional simple polytope