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Sharp Threshold of global existence for the Klein-Gordon Equations with Critical Nonlinearity

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Acta Mathematicae Applicatae Sinica 2009年第2期25卷 273-282页

作者： Zai-hui Gan Bo-ling Guo Jian Zhang College of Mathematics and Software Science Sichuan Normal University Chengdu 610068 China Institute of Applied Physics and Computational Mathematics PO Box 8009 Beijing 100088 China

In this paper, we present a cross-constrained variational method to study the Cauchy problem of the nonlinear Klein-Gordon equations with critical nonlinearity in two space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish a sharp threshold of global existence and blowup of it. Furthermore, we answer the question： How small are the initial data if the solution exists globally.

关键词： Sharp threshold Klein-Gordon equation cross-constrained variational problem global existence blowup

ON THE global existence OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE

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Acta Mathematica Scientia 2017年第4期37卷 949-964页

作者：侯飞 Department of Mathematics and IMS Nanjing UniversityNanjing 210093China

This paper is a continue work of [4, 5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term --u/（1＋t）λpu, where λ≥ 0 and μ 〉 0 are constants. We have showed that, for all λ ≥ 0 andμ 〉 0 the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initial- boundary value problem in the half space R＋^d with space dimension d = 2, 3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤λ 〈 1 when the initial data is close to its equilibrium state. In addition, exponential decay of the fluid vorticity will also be established.

关键词： compressible Euler equations initial-boundary value problem damping global existence

OPTIMAL CONDITIONS OF global existence AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION

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Acta Mathematica Scientia 2016年第3期36卷 872-880页

作者：蒋毅罗川 VC/VR Lab and Department of Mathematics Sichuan Normal University

According to the variational analysis and the potential well argument, we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore, we give its application in... 详细信息

关键词： nonlinear parabolic equation variational analysis potential well argument global existence blow up

THE global existence AND A DECAY ESTIMATE OF SOLUTIONS TO THE PHAN-THEIN-TANNER MODEL

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Acta Mathematica Scientia 2022年第3期42卷 1058-1080页

作者： Ruiying WEI Yin LI Zheng-an YAO School of Mathematics and Statistics Shaoguan UniversityShaoguan 512005China Faculty of Education Shaoguan UniversityShaoguan 512005China Department of Mathematics Sun Yat-sen UniversityGuangzhou 510275China

In this paper,we study the global existence and decay rates of strong solutions to the three dimensional compressible Phan-Thein-Tanner *** a refined energy method,we prove the global existence under the assumption that the H^(3) norm of the initial data is small,but that the higher order derivatives can be *** the initial data belong to homogeneous Sobolev spaces or homogeneous Besov spaces,we obtain the time decay rates of the solution and its higher order spatial ***,we also obtain the usual L^(p)-L^(2)(1≤p≤2)type of the decay rate without requiring that the Lpnorm of initial data is small.

关键词： Phan-Thein-Tanner model global existence time decay rates

BLOW-UP AND global existence FOR A COUPLED SYSTEM OF DEGENERATE PARABOLIC EQUATIONS IN A BOUNDED DOMAIN

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Acta Mathematica Scientia 2007年第1期27卷 92-106页

作者：穆春来胡学刚李玉环崔泽键 College of Mathematics and Physics Chongqing University Institute of Applied Mathematics Chongqing University of Posts and Telecommunications Department of Mathematics Sichuan Normal University Department of Mathematics China West Normal University

This article deals with the conditions that ensure the blow-up phenomenon or its absence for solutions of the system ut= △u^l ＋ u^p1v^q1 and vt = △v ^m ＋ u^p2 v^q2 with homogeneous Dirichlet boundary conditions. The results depend crucially on the sign of the difference p2q1 - （l -p1）（m- q2）, the initial data, and the domain Ω.

关键词： global existence blow-up degenerate parabolic system

Nonresonance and global existence of Nonlinear Elastic Wave Equations with External Forces

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Chinese Annals of Mathematics,Series B 2012年第3期33卷 437-452页

作者： Yi ZHOU Wei XU School of Mathematical Sciences Fudan UniversityShanghai 200433China

This paper establishes the global existence of classical solution to the system of homogeneous,isotropic hyperelasticity with time-independent external force,provided that the nonlinear term obeys a type of null *** authors first prove the existence and uniqueness of the stationary *** they show that the solution to the dynamical system converges to the stationary solution as time goes to infinity.

关键词： Nonlinear elasticity global existence Null condition External force

global existence and Blow Up of Solutions in a Cauchy Viscoelastic Problem

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Journal of Partial Differential Equations 2018年第3期31卷 224-236页

作者： LIU Gongwei TIAN Shuying College of Science Henan University of Technology Zhengzhou 450001 China. School of Science Wuhan University of Technology Wuhan 430070 China.

The Cauchy problem for a viscoelastic equation with nonlinear damping and source terms is considered. We establish the nonexistence result of global solutions with initial energy controlled above by a critical value by modifying the method introduced in a work by Autuori et al. in 2010. Then we establish global existence for arbitrary initial data or the energy in potential well. These improve earlier results in the literatures.

关键词： Cauchy problem viscoelastic equation global existence blow up

Local and global existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

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Acta Mathematica Sinica,English Series 2005年第4期21卷 881-892页

作者： Shuang Ping TAO Shang Bin CUI Department of Mathematics. Northwest Normal University Lanzhou 730070 P. R. China Department of Mathematics Zhongshan University Guangzhou 510275 P. R. China

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.

关键词： Kaup-Kupershmidt equation Initial value problem Solution Local existence global existence

The global existence of Solutions for a Cross-diffusion System

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Acta Mathematicae Applicatae Sinica 2005年第3期21卷 519-528页

作者： Yi Wang Department of Mathematics Capital Normal University Beijing 100037 China i Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 China

This paper is concerned with the global existence of solutions for a class of quasilinear cross-diffusion system describing two species competition under self and cross population pressure. By establishing and using more detailed interpolation results between several different Banach spaces, the global existence of solutions are proved when the self and cross diffusion rates for the first species are positive and there is no self or cross-diffusion for the second species.

关键词： Cross-diffusion system self-diffusion global existence a priori estimates