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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 200433 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

The fundamental solution of the Keldysh typeoperator

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Science china Mathematics 2009年第9期52卷 1829-1843页

作者： CHEN ShuXing School of mathematical Sciences, fudan university,{2., china 1. School of mathematical Sciences Fudan University Shanghai 200433 China

In this paper we discuss the fundamental solution of the Keldysh type operator $ L_\alpha u \triangleq \frac{{\partial ^2.u}} {{\partial x^2.}} + y\frac{{\partial ^2.u}} {{\partial y^2.}} + \alpha \frac{{\partial u}} {{\partial y}} $ , which is a basic mixed type operator different from the Tricomi operator. The fundamental solution of the Keldysh type operator with $ \alpha > - \frac2. {2. $ is obtained. It is shown that the fundamental solution for such an operator generally has stronger singularity than that for the Tricomi operator. Particularly, the fundamental solution of the Keldysh type operator with $ \alpha 2. {2. $ has to be defined by using the finite part of divergent integrals in the theory of distributions.

关键词： mixed type equation Keldysh operator fundamental solution finite part of divergent integral 35M10 35A08

Local Existence of Smooth Solutions to the FENE Dumbbell Model

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Chinese Annals of Mathematics,Series B 2012年第4期33卷 501-520页

作者： Ge YANG11School of mathematical Sciences, fudan university,{2., china. School of mathematical Sciences Fudan University Shanghai 200433 China

The author proves the local existence of smooth solutions to the finite extensible nonlinear elasticity (FENE) dumbbell model of polymeric flows in some weighted spaces if the non-dimensional parameter b > 2.

关键词： Fokker-Planck equation FENE model Degenerate parabolic equations Regularity

Plateau of α-function and c-minimal homoclinic orbits

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Science china Mathematics 2006年第7期49卷 922-931页

作者： LI Xia & CUI Xiaojun Department of Mathematics, university of Science and Technology of Suzhou, Suzhou 2.5009, china {2. research Institute, fudan university, shanghai 200433, china 1. Department of Mathematics University of Science and Technology of Suzhou Suzhou 215009 China 2. mathematical research Institute Fudan University Shanghai 200433 China

We consider the construction of the plateau of the α-function in a hyperbolic and positive definite Lagrangian system, and link the boundries of the α-function's plateau with the distribution of c-minimal homoclinic... 详细信息

关键词： α-function, β-function, plateau, Aubry sets.

Metalloproteins/metalloenzymes for the synthesis of acetyl-CoA in the Wood-Ljungdahl pathway

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Science china Chemistry 2009年第12期52卷 2071-2082页

作者： ZHU XiaoFei1 & TAN XiangShi1,2.1 Department of Chemistry,fudan university,{2. 200433,china 2.Institutes of Biomedical Sciences,fudan university,{2. 200433,china 1. Department of Chemistry Fudan University Shanghai 200433 China 2. Institutes of Biomedical Sciences Fudan University Shanghai 200433 China

This paper focuses on the group of metalloproteins/metalloenzymes in the acetyl-coenzyme A synthesis pathway of anaerobic microbes called Wood-Ljungdahl pathway,including formate dehydrogenase (FDH),corrinoid iron sulfur protein (CoFeSP),acetyl-CoA synthase (ACS) and CO dehydrogenase (CODH). FDH,a key metalloenzyme involved in the conversion of carbon dioxide to methyltetrahydrofolate,catalyzes the reversible oxidation of formate to carbon dioxide. CoFeSP,as a methyl group transformer,accepts the methyl group from CH3-H4 folate and then transfers it to ACS. CODH reversibly catalyzes the reduction of CO2.to CO and ACS functions for acetyl-coenzyme A synthesis through condensation of the methyl group,CO and coenzyme A,to finish the whole pathway. This paper introduces the structure,function and reaction mechanisms of these enzymes.

关键词： Wood-Ljungdahl pathway formate dehydrogenase corrinoid iron sulfur protein acetyl-CoA synthase CO dehydrogenase

Optimal variational principle for backward stochastic control systems associated with Lévy processes

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Science china Mathematics 2012年第4期55卷 745-761页

作者： TANG MaoNing 1 & ZHANG Qi 2.1 Department of mathematical Sciences,Huzhou university,Huzhou 313000,{2. 2.School of mathematical Sciences,fudan university,shanghai 200433,{2. 1. Department of mathematical Sciences Huzhou University Huzhou 313000 China2. School of Mathematical Sciences Fudan University Shanghai 200433 China

The paper is concerned with optimal control of backward stochastic differentiM equation （BSDE） driven by Teugel＇s martingales and an independent multi-dimensional Brownian motion, where Teugel＇s martingales are a family of pairwise strongly orthonormal martingales associated with L6vy processes （see e.g., Nualart and Schoutens＇ paper in 2.00）. We derive the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria （or backward linear-quadratic problem, or BLQ problem for short） is discussed and characterized by a stochastic Hamilton system.

关键词： stochastic control stochastic maximum principle Ldvy processes Teugel's martingales backwardstochastic differential equations

Hyperbolic positive mass theorem under modified energy condition

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Science china Mathematics 2008年第12期51卷 2189-2197页

作者： XIE NaQing Institute of Mathematics, School of mathematical Sciences, fudan university,{2., china Institute of Mathematics School of Mathematical Sciences Fudan University Shanghai 200433 China

We provide two new positive mass theorems under respective modified energy conditions allowing T00 negative on some compact set for certain modified asymptotically hyperbolic manifolds. This work is analogous to Zhang... 详细信息

关键词： positive mass theorem modified energy condition modified asymptotically hyperbolic 83C40 53C2.

THE MODULI SPACE OF COMPLEX LAGRANGIAN SUBMANIFOLDS IN THE HYPER-KAEHLER MANIFOLD

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Chinese Annals of Mathematics,Series B 1999年第4期20卷 393-400页

作者： FU JIXIANG(Institute of Mathematics, fudan university, shanghai 200433,{2.) fudan Univ Inst Math Shanghai 200433 Peoples R China

The deformation of a compact complex Lagrangian submanifold in a hyper-Kaehler manifold and the moduli space are studied. It is proved that the moduli space Mc1 is a special Kaehler manifold, where special means that there is a real flat torsionfree symplectic connection satisfying dI = 0 (I is a complex structure of Mcl). Thus, following [4], one knows thatT*Mcl is a hyper-Kaehler manifold and then that Mcl is a complex Lagrangian submanifold in T* Mcl.

关键词： Moduli space Complex Lagrangian submanifold Hyper-Kaehler manifold Special Lagrangian submanifold Special Kaehler manifold Deformation

THE OCCUPATION DENSITY FIELD FOR THE CATALYTIC SUPER-BROWNIAN MOTION

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Chinese Annals of Mathematics,Series B 1999年第4期20卷 447-454页

作者： HONG WENMING(Institute of Mathematics, fudan university shanghai 200433,{2.) fudan Univ Inst Math Shanghai 200433 Peoples R China

It is proved that the occupation time of the catalytic super-Brownian motion is absolutely continuous for d = 1, and the occupation density field is jointly continuous and jointly Holder continuous.

关键词： Branching rate functional Brownian collision local time Catalytic super-Brownian motion Occupation time Occupation density field