Some degenerate and quasilinear parabolic systems not in divergence form

This paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form: u_t=u^p(Δu+av), v_t=v^q(Δv+bu), with null Dirichlet boundary conditions and positive initial conditions, where p, q, a and b are all positive constants. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that all solutions exist globally if and only if ab?λ₁², where λ₁ is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition.     

具连续时滞Liénard方程概周期解的存在唯一性

结合Liapunov泛函，研究了具连续时滞Lienard方程概周期解的存在唯一性和安全一致渐进稳定性。     

Stability of steady nonisothermal flow in a vertical layer with permeable boundaries in the microconvection model

The problem of the stability of steady convective viscous incompressible fluid flow in a vertical layer with boundaries at different temperatures is considered in the presence of transverse blowing through the layer. The complete spectral problem is solved for a silicon melt. The neutral curve is constructed and the critical Grashof number is found. The numerical calculations show that the presence of transverse blowing significantly affects the flow stability. As compared with the Oberbeck-Boussinesq model, in the microconvection model the instability develops at lower wavenumbers.     

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

ROBUST GLOBAL EXPONENTIAL STABILITY OF UNCERTAIN IMPULSIVE SYSTEMS

By using the quasi-Lyapunov function, some sufficient conditions of global exponential stability for impulsive systems are established, which is the basis for the following discussion. Then, by employing Riccati inequality and Hamilton-Jacobi inequality approach, some sufficient conditions of robust exponential stability for uncertain linear/nonlinear impulsive systems are derived, respectively. Finally, some examples are given to illustrate the applications of the theory.     

具有开发的非自治单种群生态系统研究

研究了具有开发的单种群非自治周期系统．利用非自治微分方程理论．以及变分法理论和泛函极值的Euler方程方法,得到了该开发系统的持续生存性．周期解存在性，全局渐近稳定性．研究了在开发情况下的最大收获量问题，以及取得最大经济效益下的收获问题等．     

具有急慢性阶段的MSIS流行病模型阈值和稳定性结果

系统研究了具有急性和慢性两个阶段的MSIS流行病模型.由两节构成,第1节建立和研究了具有急慢性阶段的MSIS流行病模型;第2节在第1节的基础上建立和研究了具有慢性病病程的MSIS流行病模型.第1节的模型是四个常微分方程构成的方程组.第2节的模型既含有常微分方程,又含有偏微分方程.运用微分方程和积分方程中的理论和方法,得到了这两个模型再生数R0的表达式.证明了当R0<1时,无病平衡态是全局渐近稳定性,给出了各模型地方病平衡态的存在性和稳定性条件.     

Three-Step Predictor-Corrector of Exponential Fitting Method for Nonlinear Schroedinger Equations

We develop the three-step explicit and implicit schemes of exponential fitting methods. We use the three- step explicit exponential fitting scheme to predict an approximation, then use the three-step implicit exponential fitting scheme to correct this prediction. This combination is called the three-step predictor-corrector of exponential fitting method. The three-step predictor-corrector of exponential fitting method is applied to numerically compute the coupled nonlinear Schroedinger equation and the nonlinear Schroedinger equation with varying coefficients. The numerical results show that the scheme is highly accurate.     

Variational Derivative for Buckling Loads of Beams and Frames

The finite-difference method is a numerical technique for obtaining approximate solutions to differential equations. The main objective of the present study is to give a new aspect to the finite-difference method by using a variational derivative. By applying this formulation, accurate values of the buckling loads of beams and frames with various end supports are obtained. The performance of this formulation is verified by comparison with numerical examples in the literature __________ Published in Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 139–144, July 2005.     

纳米氧化钛-乙二醇溶胶溶液的研究

选用了钛酸丁酯、硬脂酸和乙二醇作为表面活性剂，采用表面化学修饰和表面物理修饰2种方法修饰纳米氧化钛，然后分散在乙二醇溶剂中形成溶胶溶液．并通过红外光谱仪、紫外分光计、原子力显微镜，分析了表面化学修饰后的纳米氧化钛表面化学结构的变化，观测了纳米氧化钛溶胶在乙二醇溶剂中稳定性．试验结果表明表面活性剂与纳米氧化钛的表面的不饱和键之间形成了新的化学结构，粒子表面可能接枝上有机长链，提高了纳米粒子在溶剂中的相容性．表面化学修饰后的纳米氧化钛与乙二醇溶剂形成了较稳定的溶胶体系，而且纳米溶胶粒径较小．表面活性剂添加量与纳米粒子添加量控制在(1～1．2)：1时，可以获得纳米溶胶粒径较小，同时溶胶稳定性较好的纳米氧化钛-乙二醇溶胶体系．