一类具阻尼项的四阶非线性波动方程的Cauchy问题

本文利用压缩映像原理和积分估计研究一类具阻尼项的四阶非线性波动方程的Cauchy问题小振幅解的整体存在性、唯一性和衰减性.     

4阶非线性薛定谔方程解的渐近行为（英文）

本文综述了具有幂次非线性或耗散形式的非线性4阶薛定谔方程小振幅解的全局存在性和渐近行为的一些最新研究结果.考虑了散射问题的超临界非线性和临界非线性情况.     

非线性项依赖于时间和空间变量的梁方程拟周期解的存在性

本文研究带有空间周期和时间拟周期非线性项的常数势能梁方程,证明了对于大多数频率向量和大多数势能常数,方程存在小振幅、线性稳定的时间拟周期解.通过对本质上无穷多个小除数的测度估计,本文构建了一个实解析的辛坐标变换,将Hamilton函数化为其Birkhoff标准型.利用一个无穷维Kolmogorov-Arnold-Moser(KAM)定理,本文证明了拟周期解的存在性.     

一类具阻尼项的Klein Gordon方程解的衰减行为

讨论了一类带阻尼项的n维Klein Gordon方程的柯西问题,观察到其线性方程的耗散结构是正则耗散型,这将意味着在对初值的正则假设下,可以得到解的最佳衰减估计.基于其相应线性方程的衰减估计和小初值条件,利用压缩映射原理,在Sobolev空间中证明了整体解的存在性和小振幅解的渐近行为.     

具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔

研究了具有时滞耦合的n个van derPol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2:5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性,对系统在2:5共振点附近的动力学行为进行了开折和分类.得到了一些有趣的动力学现象,如振幅死区、周期解和双稳态解等,相应的数值模拟验证了理论结果的正确性.     

具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔

研究了具有时滞耦合的n个van der Pol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2∶5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性,对系统在2∶5共振点附近的动力学行为进行了开折和分类.得到了一些有趣的动力学现象,如振幅死区、周期解和双稳态解等,相应的数值模拟验证了理论结果的正确性.     

Liénard系统的比较定理

研究了Liénard方程的一类新的等价系统解的有界性与周期解的存在性.证明了几个比较定理,使传统Liénard方程等价系统解的有界性和周期解的存在性可用于判定新等价系统解的有界性与周期解的存在性.     

各项异性椭圆方程基本解的存在性

证明了右端可测的各项异性椭圆方程基本解的存在性,其中应用了各项异性Sobolev空间和Lebesgue空间.首先得到近似方程的解,然后通过对这些解的子列取极限,得到原方程的解.关键是要有一个近似函数空间以及近似方程的先验估计.最后运用Vitali定理证明了原方程基本解的存在性,推广和改进了已有方程.     

抽象空间中非线性算子方程变号解的存在性研究

利用Banach空间中的锥理论和不动点定理讨论了非线性算子方程变号解的存在性,给出了E_u_0空间下非线性算子方程变号解至少有一个变号解、一个正解和一个负解的条件,并讨论了仅通过一个上解条件得出非线性算子方程变号解的存在性定理.     

关于一类高阶Liénard型方程周期解的注记

本文研究一类具偏差变元的高阶 Liénard型方程的周期解存在性 ,给出了这类方程周期解存在性的若干充分条件 .     

Travelling waves of a predator–prey model with a spatiotemporal delay

This paper is concerned with the existence of traveling waves to a predator–prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper–lower solutions and by using the cross‐iteration method as well as the Schauder's fixed‐point theorem, the existence of a traveling wave solution connecting the semi‐trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Travelling waves of a delayed SIRS epidemic model with spatial diffusion

This paper is concerned with the existence of travelling waves to an SIRS epidemic model with bilinear incidence rate, spatial diffusion and time delay. By analysing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder’s fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

有界域上三维非等熵半导体方程解的渐近性(英文)

本文讨论了一类三维非等熵半导体方程. 利用能量估计法, 证明了热平衡稳态解的存在唯一性.然后, 得到了Cauchy-Neumann问题光滑解的整体存在性以及当t→ +∞这种光滑解以指数速度收敛到稳定解, 改进了文献[12]的结果.     

Travelling waves in an infectious disease model with a fixed latent period and a spatio–temporal delay

This paper is concerned with the existence of travelling waves to an infectious disease model with a fixed latent period and a spatio–temporal delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this model is discussed. By constructing a pair of upper–lower solutions, we use the cross iteration method and the Schauder’s fixed point theorem to prove the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Steady state analysis of a nonlinear renewal equation

In the analysis of a nonlinear renewal equation it is natural to anticipate the existence of nonzero steady states and deal with the question of their stability. Sufficient conditions for existence and uniqueness for these steady states are given. The study of the linearized version of the renewal equation around the steady state helps to a great extent to have insight into some complicated dynamics of the full problem. At this stage the first eigenvalue of the steady state plays a vital role. The characteristic equation, a functional equation whose roots are the eigenvalues, is derived. We give various structures showing that the steady state may be stable or unstable (though the fertility rate is decreasing with competition). A similar study is carried out on a nonlinear model motivated by neuroscience in which the total population is conserved.     

Stability of a stochastic SEIS model with saturation incidence and latent period

In this paper, a stochastic SEIS epidemic model with a saturation incidence rate and a time delay describing the latent period of the disease is investigated. The model inherits the endemic steady state from its corresponding deterministic counterpart. We first show the existence and uniqueness of the global positive solution of the model. Then, by constructing Lyapunov functionals, we derive sufficient conditions ensuring the stochastic stability of the endemic steady state. Numerical simulations are carried out to confirm our analytical results. Furthermore, our simulation results shows that the existence of noise and delay may cause the endemic steady state to be unstable.     

具有接种和非局部扩散的流行病模型的行波解（英文）

An epidemic model with vaccination and nonlocal diffusion is proposed, and the existence of traveling wave solutions of this model is studied. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder’s fixed point theorem,sufficient conditions are obtained for the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.     

Non-constant positive steady state of one resource and two consumers model with diffusion

In this paper, a predator-prey reaction-diffusion system with one resource and two consumers is considered. Assume that one consumer species exhibits Holling II functional response while the other consumer species exhibits Beddington-DeAngelis functional response, and they compete for the common resource. First, it is proved that the unique positive constant steady state is stable for the ODE system and the reaction-diffusion system. Second, a prior estimates of positive steady state is given. Finally, the non-existence of non-constant positive steady state, the existence and bifurcation of non-constant positive steady state are studied.     

Steady states and dynamics of an autocatalytic chemical reaction model with decay

The dynamics and steady state solutions of an autocatalytic chemical reaction model with decay in the catalyst are considered. Nonexistence and existence of nontrivial steady state solutions are shown by using energy estimates, upper–lower solution method, and bifurcation theory. The effects of decay order, decay rate and diffusion rates to the dynamical behavior are discussed.     

具有Holling-II型功能反应和交叉扩散的捕食-食饵模型正平衡解的存在性和不存在性

本文主要研究一类在齐次Dirichlet边界条件下带交叉扩散的Holling-II型捕食者-食饵模型正平衡解的存在性, 其中两个交叉扩散系数分别代表食饵远离捕食者的趋势和捕食者追逐食饵的趋势. 应用不动点指标理论得到了正平衡解存在的充分条件, 并进一步研究了正平衡解不存在的条件.