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cauchy　problem　FOR　A　NONLINEAR　SINGULAR　INTEGRAL－DIFFERENTIAL　EVOLUTION　SYSTEM

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Applied Mathematics(A Journal of Chinese Universities) 1994年第1期9卷 45-53页

作者： ZHANG LINGHAI(Inst.of Appl.Phys.and Comp.Math.,Beijing 100088) Inst. of Appl. Phys. and Comp. Math. Beijing

We study the cauchy problem for a nonlinear evolution system with singularintegral differential *** means of some a priori estimates of the solution and theLeray-Schauder's fixed point theorem, we prove the existence ... 详细信息

关键词： cauchy problem Nonlinear Evolution System Singular integral-differentialTerm Integral Estimate.

Variational regularization method of solving the cauchy problem for Laplace's equation: Innovation of the Grad–Shafranov(GS) reconstruction

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Chinese Physics B 2014年第10期23卷 650-655页

作者：颜冰黄思训 Institute of Meteorology and Oceanography PLA University of Science and Technology State Key Laboratory of Satellite Ocean Environment Dynamics Second Institute of Oceanography State Oceanic Administration

The simplified linear model of Grad-Shafranov （GS） reconstruction can be reformulated into an inverse boundary value problem of Laplace＇s equation. Therefore, in this paper we focus on the method of solving the inverse boundary value problem of Laplace＇s equation. In the first place, the variational regularization method is used to deal with the ill- posedness of the cauchy problem for Laplace＇s equation. Then, the ＇L-Curve＇ principle is suggested to be adopted in choosing the optimal regularization parameter. Finally, a numerical experiment is implemented with a section of Neumann and Dirichlet boundary conditions with observation errors. The results well converge to the exact solution of the problem, which proves the efficiency and robustness of the proposed method. When the order of observation error δ is 10-1, the order of the approximate result error can reach 10-3.

关键词： Grad-Shafranov reconstruction variational regularization method cauchy problem

ON THE cauchy problem FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM

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Acta Mathematica Scientia 2024年第5期44卷 1766-1786页

作者： Xiao SU Shubin WANG College of Science Henan University of TechnologyZhengzhou450001China School of Mathematics and Statistics Zhengzhou UniversityZhengzhou450001China

This paper is devoted to the cauchy problem for the generalized damped Boussinesq equation with a nonlinear source term in the natural energy *** the help of linear time-space estimates,we establish the local existence and uniqueness of solutions by means of the contraction mapping *** global existence and blow-up of the solutions at both subcritical and critical initial energy levels are ***,we construct the sufficient conditions of finite time blow-up of the solutions with arbitrary positive initial energy.

关键词： damped Boussinesq equation cauchy problem global solutions blow-up

BLOW UP OF SOLUTIONS TO THE cauchy problem FOR NONLINEAR WAVE EQUATIONS

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Chinese Annals of Mathematics,Series B 2001年第3期22卷 275-280页

作者： ZHOU YI Institute of Mathematics, Fudan University, Shanghai 200433, China Institute of Mathematics Fudan University Shanghai 200433 China

The author proves blow up of solutions to the cauchy problem of certain nonlinear equations and, also, estimates the time when the blow up occurs.

关键词： cauchy problem Nonlinear wave equation Blow up

The cauchy problem of the Nonlinear Schr■dinger Equations in R^(1+1)

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Acta Mathematicae Applicatae Sinica 2007年第4期23卷 593-610页

作者： Gui-xiang Xu Institute of Applied Physics and Computational Mathematics P.O. Box 8009 Beijing 100088 China

In this paper, we consider the local and global solution for the nonlinear Schrodinger equation with data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data. The techniques to be used are adapted from the Strichartz type estimate, Kato＇s smoothing effect and the maximal function （in time） estimate for the free SchrSdinger operator.

关键词： Schrodinger equation cauchy problem besov space littlewood-paley decomposition self-similar solution

ON THE STABILITY OF THE POSITIVE RADIAL STEADY STATES FOR A SEMILINEAR cauchy problem INVOLVING CRITICAL EXPONENTS

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Acta Mathematica Scientia 2008年第2期28卷 348-354页

作者：邓引斌杨芬 Department of Mathematics Huazhong Normal University Department of Information and Computation Science College of Science Wuhan University of Science and Technology

This article is contributed to the cauchy problem {δu/δt=△u＋K（｜x｜）u^p in R^n×（0,T）, u（x,0）=φ（x） in R^n;with initial function φ≡/0. The stability of positive radial steady state, which are positive so... 详细信息

关键词： Stability cauchy problem asymptotic stability

Lipschitz Continuous Solutions to the cauchy problem for Quasi-linear Hyperbolic Systems

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Chinese Annals of Mathematics,Series B 2012年第4期33卷 521-536页

作者： Xiang CHEN School of Mathematical Sciences Fudan University Shanghai 200433 China.

Lipschitz continuous solutions to the cauchy problem for 1-D first order quasi-linear hyperbolic systems are considered. Based on the methods of approximation and integral equations, the author gives two definitions o... 详细信息

关键词： First order quasi-linear hyperbolic systems Lipschitz continuous solution cauchy problem Existence and uniqueness

Well-Posedness of the cauchy problem for the Coupled System of the Schrodinger-KdV Equations

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Acta Mathematica Sinica,English Series 1999年第2期15卷 215-224页

作者： Boling Guo Changxing Miao, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, P. R. ChinaFax: 86-10-6201018 E-mail: js8s@*** Institute of Applied Physics and Computational Mathematics Beijing P. R. China

This paper is devoted to the study of the cauchy problem for the coupled system of the Schrodinger-KdV equations which describes the nonlinear dynamics of the one-dimensional Langmuir and ion-acoustic waves. Global well-posedness of the problem is established in the space H^k×H^k (k∈ Z^+), the first and second components of which correspond to the electric field of the Langmuir oscillations and the low-frequency density perturbation respectively.

关键词： cauchy problem Schrodinger-KdV equations Well-posedness

SHARP WELL-POSEDNESS OF THE cauchy problem FOR THE HIGHER-ORDER DISPERSIVE EQUATION

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Acta Mathematica Scientia 2017年第4期37卷 1061-1082页

作者：蒋敏杰闫威张贻民 College of Mathematics and Information Science Henan Normal University School of Science Wuhan University of Technology

This current paper is devoted to the cauchy problem for higher order dispersive equation ut＋δx^2n＋1u=δx（uδx^nu）＋δx^n-1（ux^2）, n ≥ 2, n ∈ N^＋. Ut By using Besov-type spaces, we prove that the associated problem is locally well-posed in H（-n/2＋3/4,-1/2n）. The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H^（s,a）（R） with s〈-n/2＋3/4 and all a∈R.

关键词： cauchy problem sharp well-posedness modified Bourgain spaces