正阻尼摆方程的全局吸引性

本文考虑摆方程d~2x/dt~2+f(t,x)dx/dt+g(t,x)=0,这里f(t,x),g(t,x)∈C~1(R~1/ωZ~1×R~1/ω_1Z~1).我们给出一个充分条件保证上方程在其轨道空间(t,x,x)∈R~1/ωZ~1×R~1/ω_1Z~1×R~1中有唯一的不变环面,且方程的所有解都指数趋于此不变环面.     

正阻尼摆方程的全局吸引性

本文考虑摆方程d~2x/dt~2+f(t,x)dx/dt+g(t,x)=0,这里f(t,x),g(t,x)∈C~1(R~1/ωZ~1×R~1/ω_1Z~1).我们给出一个充分条件保证上方程在其轨道空间(t,x,x)∈R~1/ωZ~1×R~1/ω_1Z~1×R~1中有唯一的不变环面,且方程的所有解都指数趋于此不变环面.     

无界区域R3上的非线性应变波方程与薛定谔方程藕合方程组的指数吸引子

本文研究了无界区域R~3上的非线性应变波方程与薛定谔方程藕合方程组.运用带权空间构造一类紧算子和算子分解的方法,得到了该方程在无界区域R3上的H~2×H~2×H~1(R~3)中拥有一个指数吸引子.     

一类非线性Schrdinger方程的解

本文讨论这样一类非线性Schrdinger方程:其中k,v是常数,v>O;Ω■R~2;u(x,t)是复值未知函数;u_0是初值,已知。 给出1.方程整体解的存在性和解的估计;2.在Ω是R~2中的外区域情况下,当t→∞时解的渐近性。     

一类Schr?dinger-Hartree方程爆破解的门槛条件

该文在R~3中研究如下Schr?dinger-Hartree方程i?_tψ+△ψ=-(|x|~(-1)*|ψ|~α)|ψ|~(α-2)ψ,t0,x∈R~3,α≥2.(P)利用Gagliardo-Nirenberg与方程(P)的质量守恒律,能量守恒律建立方程的发展不变流.以此为基础在7/3≤α5时,得到其Cauchy问题的爆破解和整体解的门槛条件.     

有界区域上非线性 Schrodinger 方程解的全局性质

本文解决了 H.Bréis 在文[1]中提出的一个问题,对于非线性 Schr(?)dinger 方程(?) (1)H.Brézis 在文[1]中证明了,如果 Ω=R~2,p=3,K≤0或k>0,同时K integral from R~2|u_0|~2dx<4,(1)式有解 u(x,t),u∈C~1([0,∞),L~2(R~2))∩C([0,∞),H~2(R~2)).并且指出,在 Ω=R~2的情形,如果 p≥3,并且初值 u_0满足     

有界区域上非线性Schrdinger方程解的全局性质

本文解决了 H.Bréis 在文[1]中提出的一个问题,对于非线性 Schr(?)dinger 方程(?) (1)H.Brézis 在文[1]中证明了,如果 Ω=R~2,p=3,K≤0或k>0,同时K integral from R~2|u_0|~2dx<4,(1)式有解 u(x,t),u∈C~1([0,∞),L~2(R~2))∩C([0,∞),H~2(R~2)).并且指出,在 Ω=R~2的情形,如果 p≥3,并且初值 u_0满足     

耗散MKdV方程的指数吸引子

讨论了无界区域R~1上的MKdV方程,运用带权空间构造一类紧算子和算子分解的方法,得到该方程在H~2（R~1）上指数吸引子的存在性.     

R~3上分数阶Kirchhoff方程正解的多重性及集中性

本文考虑如下分数阶Kirchhoff方程:{M(∫∫R~3×R~3|u(x)-u(y)|~2|x-y|3+2sdxdy)(-?)su(x)+V (x)u=f (u), x∈R~3,u∈H~s(R~3),其中M(t)=ε~(2s)a+ε~(4s-3)bt是Kirchhoff函数,3/4s1,ε0是小参数,位势V是正连续函数且有全局极小,非线性项f连续且在无穷远处次临界增长.利用Ljusternik-Schnirelmann畴数理论,本文得到了正解个数与位势V全局极小集拓扑之间的关系,证明了当ε→0~+时,这些正解在H~s(R~3)中收敛到极限方程的基态解,且这些解集中在位势V的全局极小附近.此外也得到了解的衰减估计.     

非线性薛定谔-麦克斯韦方程基态解的存在性

对V,K和f作出一些假设,用山路定理得出如下的薛定谔-麦克斯韦方程基态解:{-Δu+V(x)u+K(x)φu=f(x,u), in R~3,-Δφ=K(x)u~2,in R~3.(*)     

E-紧框架小波来自MRA的特征刻画

设E=■或■,■(x)∈L~2(R~2)且■_(jk)(x)=2■(E~jx-k),其中j∈Z,k∈Z~2.若{■_(jk)|jJ∈Z,k∈Z~2}构成L~2(R~2)的紧框架,则称■(x)为E-紧框架小波.本文给出E-紧框架小波是MRA E-紧框架小波的一个充要条件,即E紧框架小波■来自多尺度分析当且仅当线性空间F_■(ξ)的维数为0或1,其中F_■(ξ)=■(ξ)|j■1},■_j(ξ)={■((E~T)~j(ξ+2kπ))}_(k∈EZ~2,j■1。     

EXISTENCE CONDITIONS OF A SPECIAL TYPE OF LIAPUNOV FUNCTIONAL FOR TWO-DIMENSIONAL DELAY SYSTEM

In this paper, the author considers the two-dimensional delay systems $$\[\mathop x\limits^ \cdot (t) = Ax(t) + Bx(t - r),A,B \in {R^{2 \times 2}},x \in {R^2},r = const \ge 0\]$$ and gives the necessary and sifficient conditions under which where exists a simple type of positive definite Liapunov functional $$\[V(\varphi ) \buildrel \Delta \over = {\varphi ^''}(0){T_\varphi }(0) + \int_{ - \tau }^0 {{\varphi ^''}(\theta )E\varphi (\theta )d\theta } \]$$ and $\[\alpha (s)\]$(where T , E are positive definite 2x2 matrices, $\[\varphi \in C([ - \tau ,0],{R^n})\]$, "." stands for transpose, $\[\alpha (s)\]$ is continuous and $\[\alpha (0) = 0,\alpha (s) > 0,s > 0\]$. such that $\[{V_{(*)}}(\varphi ) \le - \alpha (\left| {\varphi (0)} \right|).\]$.     

The Perron-Frobenius theorem for homogeneous, monotone functions

If is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that has an eigenvector in the positive cone, . We associate a directed graph to any homogeneous, monotone function, , and show that if the graph is strongly connected, then has a (nonlinear) eigenvector in . Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is ``really' about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.

     

带衰退记忆的经典反应扩散方程的全局吸引子

当非线性项满足任意阶多项式增长且外力项仅属于H~(-1)(Ω)时,研究了带衰退记忆的经典反应扩散方程的长时间动力学行为.应用抽象函数理论、半群理论以及新的估计技巧,在空间L~2(Ω)×L_μ~2(R~+;H_0~1(Ω))上证明了全局吸引子的存在性.该结果改进和推广了Chepyzhov等人(2006)及Zhong等人(2006)的相应结果.     

On the Global Well-Posedness of 3-D Boussinesq System with Variable Viscosity

In this paper, the authors first consider the global well-posedness of 3-D Boussinesq system, which has variable kinematic viscosity yet without thermal conductivity and buoyancy force, provided that the viscosity coefficient is sufficiently close to some positive constant in L∞and the initial velocity is small enough in B_(3,1)~0(R~3). With some thermal conductivity in the temperature equation and with linear buoyancy force θe3 on the velocity equation in the Boussinesq system, the authors also prove the global well-posedness of such system with initial temperature and initial velocity being sufficiently small in L~1(R~3)and B_(3,1)~0(R~3) respectively.     

Instanton counting on blowup. II. K-theoretic partition function

We study Nekrasov's deformed partition function $Z(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta)$ of 5-dimensional supersymmetric Yang-Mills theory compactified on a circle. Mathematically it is the generating function of the characters of the coordinate rings of the moduli spaces of instantons on $\mathbb R^4$. We show that it satisfies a system of functional equations, called blowup equations, whose solution is unique. As applications, we prove (a) $F(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta) = \varepsilon_1\varepsilon_2 \log Z(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta)$ is regular at $\varepsilon_1 = \varepsilon_2 = 0$ (a part of Nekrasov's conjecture), and (b) the genus $1$ parts, which are first several Taylor coefficients of $F(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta)$, are written explicitly in terms of $\tau = d^2 F(0,0,\vec{a};\mathfrak q,\boldsymbol\beta)/da^2$ in rank $2$ case.     

The Maximal Riesz, Fejér, and Cesàro Operators on Real Hardy Spaces

We prove that the maximal Riesz operator $\sigma^{\alpha,\gamma}_*$ is of strong type from $L^1(\R) \cap H^p$ $ (\R)$ to $L^p (\R)$ for $\alpha, \gamma>0$ and $1/(1+\alpha) < p \le 1$, it is of weak type for $\alpha,\gamma>0$ and $1/(1+\alpha) = p$, and these results are best possible. The proofs are based on sharp estimates of the derivatives of the Riesz kernel. We characterize the real Hardy space $H^p(\R)$ in terms of $\sigma^{\alpha,1}_*$ for $1/(1+ \alpha) < p \le 1$, and draw consequences for real Hardy spaces on $\R^2$, as well. For example, an integrable function $f$ belongs to $H^1(\R)$ if and only if the maximal Fej\er operator $\sigma^{1,1}_*$ applied to $f$ belongs to $L^1(\R)$. We also establish analogous results for real Hardy spaces on $\T$ and $\T^2$.     

WAVELET DECOMPOSITIONS IN $L^2(R^2)$

Let ${V_k}^{+\infinity}_{k=-\infinity}$ be a multiresolution analysis generated by a function $\phi(x)\in L^2(R^2)$. Under this multiresolution framework the key point for studying wavelet decompositions in $L^2(R^2)$ is to study the properties of Wo which is the orthogonal complement of $V_0$ in $V_1:V_1=V_0\oplus W_0$.In this paper the author studies the structure of W_0 and furthermore shows that a box spline of three directions can generate a wavelet decomposition of $L^2(R^2)$.     

Convolution with affine arclength measures in the plane

We obtain an estimate for the norm of a certain convolution operator.

     

带有衰退记忆的非自治经典反应扩散方程的吸引子

当非线性项以任意阶多项式增长,且外力项仅为平移有界而非平移紧时,研究了非自治经典反应扩散方程解的长时间动力学行为.应用抽象函数理论和新的估计技术,在拓扑空间L~2(Ω)×L_μ~2(R~+;H_0~1(Ω)上,证明了一致吸引子的存在性.该结果改进和推广了Chepyzhov等(2006)和钟承奎等(2006)的相应结果.