GLOBAL WELL-POSEDNESS FOR THE KLEIN-GORDON EQUATION BELOW THE ENERGY NORM

We study global well-posedness below the energy norm of the Cauchyproblem for the Klein-Gordon equation in R^n with n≥3. By means of Bourgain‘s method along with the endpoint Strichartz estimates of Keel and Tao, we prove the H^s-global well-posedness with s<1 of the Cauchy problem for the Klein-Gordon equation. This we do by establishing a series of nonlinear a priori estimates in the setting of Besov spaces.     

GLOBAL EXISTENCE OF SOLUTIONS FOR QUADRATIC QUASI-LINEAR KLEIN-GORDON SYSTEMS IN ONE SPACE DIMENSION

Consider quadratic quasi-linear Klein-Gordon systems with eventually different masses for small, smooth, compactly supported Cauchy data in one space dimension.It is proved that the global existence holds when a convenient null condition is satisfied by nonlinearities.     

Sine—Gordon方程的全局吸引子的维数估计

本文得到了阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的狄氏问题的全局吸引子的Ｈａｕｓｄｏｒｆｆ维数以偶数上界的参数条件，特别地，当阻尼与Ｌａｐｌａｅ算子的第一个特征值适当大时，全局吸引子是零维的，零维吸引子恰是系统的唯一平衡解并且指数吸引相空间的有界集。     

GLOBAL WELL-POSEDNESS AND SCATTERING FOR THE MASS-CRITICAL HARTREE EQUATION IN HIGH DIMENSIONS

We obtain global well-posedness and scattering, and global Lt,x2(d+2)/d spacetime bounds for solutions to the defocusing mass-critical Hartree equation in Rt × Rxd, d ≥ 5.     

三维古地温度场方程的参数优化辨识

为较真实地反映沉积盆地古地温的变化情况，本文建立了适用于不同岩性的沉积盆地地层的古地温度场方程及其定解条件；依据优化理论构造了地层物性参数的优化识别模型，并给出了该优化识别问题最优解存在的必要条件和最优化算法。     

半线性抛物方程在L2p空间中全局吸引子的存在性

本文研究了半线性抛物方程所生成的半群{S（t）}t≥0的吸引子的存在性.利用文献[1]中证明吸引子正则性的思想,分别得到半群{S（t）}t≥0在L^2p（Ω）空间中具有一个有界吸收集和一个全局吸引子.     

EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE Hk

In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k（Ω） in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞）.     

GLOBAL EXISTENCE OF SOLUTIONS OF CAUCHY PROBLEM FOR GENERALIZED BBM-BURGERS EQUATION

In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation （1） are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.     

GLOBAL STRONG SOLUTIONS FOR A CLASS OF BBM-BO TYPE EQUATION

1IntroductionBBMequatiollbut box~'uxxt=0,x6R,t>0,(l.1)isderivedinnonlineardispersivesystemsbyT.B.Benjamin,J.H.L.BonaandJ.J.Mahony'asamodelequationforlongwaves.TheypointedoutthatthismodelismoresuitabletoreplaceKDVequation(see[2],[4]).BOequationILt 2'UU. HU.:.=0,XER,t>0,(1.2)arisesinthestudyoflollginternalgravitywavesilldeepstrafiedfluids(see[51,[9]).Bothofthelllarewell-kllownandhavebeenwidelystudied(see[31,[10],[11],[12]).Inthispaper,wecollsiderthefollowingBBM-BOtypeequationwithini…     

GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR GENERALIZED QUANTUM MHD EQUATION

We prove the existence of a weak solution for a generalized quantum MHD equation in a 2-dimensional periodic box for large initial data. The existence of a global weak solution is established through a three-level approximation,energy estimates, and weak convergence for the adiabatic exponent γ 1.     

全空间上一类非齐次椭圆方程正解的衰减

本文研究了非齐次椭圆方程不同衰减正解的存在性和衰减问题.利用上下解方法,得到了两个有着不同衰减的正解的存在性,推广了文献[2]中非齐次方程正解的存在性.     

GLOBAL WELL-POSEDNESS FOR A FIFTH-ORDER SHALLOW WATER EQUATION ON THE CIRCLE

The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u + 3 x u 5 x u + 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s (T) for s > 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.     

GLOBAL EXISTENCE FOR A CLASS OF SYSTEMSOF NONLINEAR WAVE EQUATIONS INTHREE SPACE DIMENSIONS

Consider a system of nonlinear wave equationsfor i = 1, … , m, where F, (i = 1, … , m) are smooth functions of degree 2 near the origin of their arguments, and u = (u1, … ,um), while u and x u represent the first and second derivatives of u, respectively. In this paper, the author presents a new class of nonlineaxity for which the global existence of small solutions is ensured. For example, global existence of small solutions for arbitrary cubic terms,arbitrary cubic termswill be established, provided that c12 ≠ c22.     

高维弱阻尼Schr dinger方程的渐近行为

本文研究R3中有界区域Ω上具有零阶耗散和外力驱动的非线性Schrdinger方程.我们首先得到整体解的存在唯一性结果,然后证明其长时间行为由V=H1(Ω)中紧的整体吸引子刻划.     

PIECEWISE SMOOTH SOLUTIONS OF SEMILINEAR HYPEREOLIC SYSTEMS IN HIGHER SPACE DIMENSION

This paper discusses piecewise smooth solutions for semilinear hyperbolic systems in multi-dimensional space. In the class of piecewise smooth functions the author proves the existence and uniqueness of local solutions to the Cauchy problem for $\[3 \times 3\]$ hyperbolic system. Besides， it is also proved that when two characteristic surfaces bearing weak singularities intersect, the solution will still be piecewise smooth and the weak singularities will propagatealong all characteristic surfaces.     

WELL-POSEDNESS AND SPACE-TIME REGULARITY OF SOLUTIONS TO THE LIQUID CRYSTAL EQUATIONS IN CRITICAL SPACE

In this paper,we consider a hydrodynamic flow of nematic liquid crystal system.We prove the local well-posedness for the system in the critical Lebesgue space,and study the space-time regularity of the local solution.     

WELL-POSEDNESS AND SPACE-TIME REGULARITY OF THE SOLUTIONS TO THE LIQUID CRYSTAL EQUATIONS IN CRITICAL SPACE

In this paper, we consider a hydrodynamic flow of nematic liquid crystal system. We prove the local well-posedness for the system in the critical Lebesgue space, and study the space-time regularity of the local solution.     

GLOBAL EXISTENCE AND ASYMPTOTICS BEHAVIOR OF SOLUTIONS FOR A RESONANT KLEIN-GORDON SYSTEM IN TWO SPACE DIMENSIONS

The authors study a resonant Klein-Gordon system with convenient nonlinearities in two space dimensions, prove that such a system has global solutions for small, smooth, compactly supported Cauchy data, and find that the asymptotic profile of the solution is quite different from that of the free solution.     

高维非线性四阶抛物型方程初值问题全局解

对一类高维非线性四阶抛物型方程初值问题,在初始数据适当小及非线性项适当光滑的情况下,获得了其古典解全局存在性。