Global dynamics and zero-diffusion limit of a parabolic–elliptic–parabolic system for ion transport networks

This paper is concerned with a parabolic–elliptic–parabolic system arising from ion transport networks. It shows that for any properly regular initial data, the corresponding initial–boundary value problem associated with Neumann–Dirichlet boundary conditions possesses a global classical solution in one-dimensional setting, which is uniformly bounded and converges to a trivial steady state, either in infinite time with a time-decay rate or in finite time. Moreover, by taking the zero-diffusion limit of the third equation of the problem, the global weak solution of its partially diffusive counterpart is established and the explicit convergence rate of the solution of the fully diffusive problem toward the solution of the partially diffusive counterpart, as the diffusivity tends to zero, is obtained.     

三维与二维晶体生长控制方程的精确解

本文通过将未知函数展开成复数形式的Fourier级数,求出了一类二阶偏微分方程的三角级数形式的解析解,并严格证明了其收敛性．三维稳态与二维稳态和二维非稳态晶体生长控制方程都是这类二阶偏微分方程特例．利用这一特点,本文求出了三维稳态与二维稳态和二维非稳态晶体生长控制方程的解析解．理论结果有助于揭示稳态晶体生长的本质特性．本文还给出了三维非稳态晶体生长控制方程的解析解．     

Exact controllability for the magnetohydrodynamic equations

We study the local exact controllability of the steady state solutions of the magnetohydrodynamic equations. The main result of the paper asserts that the steady state solutions of these equations are locally controllable if they are smooth enough. We reduce the local exact controllability of the steady state solutions of the magnetohydrodynamic equations to the global exact controllability of the null solution of the linearized magnetohydrodynamic system via a fixed‐point argument. The treatment of the reduced problem relies on two Carleman‐type inequalities for the backward adjoint system. © 2003 Wiley Periodicals, Inc.     

Exact solution for the pulsating finite gap Dean flow

Analytical solutions of the Navier–Stokes equations for the fully developed laminar flow in a cylindrical annulus, when an oscillating circumferential pressure gradient is imposed (finite gap oscillating Dean flow), are presented. The solution for the case of steady flow, which has been given by Goldstein, is obtained as a limit case of the oscillating flow when the frequency of the oscillating pressure gradient tends to zero. The pulsating flow solution is obtained by the superposition of the constant and oscillating pressure gradient solutions.     

Exact multiplicity of solutions and S-shaped bifurcation curve for a class of semilinear elliptic equations

The set of steady state solutions to a reaction-diffusion equation modeling an autocatalytic chemical reaction is completely determined, when the reactor has spherical geometry, and the spatial dimension is n=1 or 2 for any reaction order, or n?3 for subcritical reaction order. Bifurcation approach and analysis of linearized problems are used to establish exact multiplicity and precise global bifurcation diagram of positive steady states.     

Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions

The existence of global smooth solutions to the multidimensional hydrodynamic model for plasmas of electrons and positively charged ions with insulating boundary conditions is shown under the assumption that the initial densities are close to a constant. Furthermore it is proved that the particle densities converge exponentially fast to the constant steady state.     

Examples of non-uniqueness for the linear problem of potential flow around semi-submerged bodies

The plane Neumann-Kelvin problem, which uses a linear approximation of the theory of waves of small amplitude to describe the steady, vortex-free motion of semi-submerged cylindrical bodies in an ideal, incompressible, heavy liquid with a free surface, is considered. For each fixed value of the free-stream velocity and a convoy of two or more bodies it is shown that the geometry of the bodies can be so chosen that the homogeneous Neumann-Kelvin problem will have a non-trivial solution. A family of potentials is constructed that provide such examples of non-uniqueness. The corresponding configurations can be obtained by choosing some of the streamlines of the solutions as the body contours. Examples are given.     

带扩散项单种群模型的精确能控性

本文研究了一类单种群生态模型的精确能控性问题.利用线性化系统的Carleman估计和Kakutani不动点定理,证明了该模型稳态解的局部能控性.它反应了外部控制在种群演变过程中的影响作用.     

Large Amplitude Steady Periodic Waves for Fixed-Depth Rotational Flows

We consider steady periodic water waves with vorticity which propagate over a flat bed with a specified fixed mean-depth d > 0. Following a novel reformulation of the governing equations, we use global bifurcation theory to establish a global continuum of solutions throughout which the mean-depth is a fixed quantity. Furthermore, we establish the limiting behavior of solutions in this continuum, which include the existence of weak stagnation points, that are characteristic of large-amplitude steady periodic water waves.     

Global stability of solutions to a free boundary problem of ductal carcinoma in situ

In the paper we present some remarks on the global stability of steady state solutions to a free boundary model studied by Xu (2004) and also prove some new results of global stability of steady state solutions to the model.     

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.     

Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system

A diffusive predator-prey system with Holling type-II predator functional response subject to Neumann boundary conditions is considered. Hopf and steady state bifurcation analysis are carried out in details. In particular we show the existence of multiple spatially non-homogeneous periodic orbits while the system parameters are all spatially homogeneous. Our results and global bifurcation theory also suggest the existence of loops of spatially non-homogeneous periodic orbits and steady state solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation.     

Global bifurcation and stability of steady states for a bacterial colony model with density-suppressed motility

We investigate the structure and stability of the steady states for a bacterial colony model with density-suppressed motility. We treat the growth rate of bacteria as a bifurcation parameter to explore the local and global structure of the steady states. Relying on asymptotic analysis and the theory of Fredholm solvability, we derive the second-order approximate expression of the steady states. We analytically establish the stability criterion of the bifurcation solutions, and show that sufficiently large growth rate of bacteria leads to a stable uniform steady state. While the growth rate of bacteria is less than some certain value, there is pattern formation with the admissible wave mode. All the analytical results are corroborated by numerical simulations from different stages.     

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.     

Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case

We prove the non-existence of non-constant positive steady state solutions of two reaction-diffusion predator-prey models with Holling type-II functional response when the interaction between the predator and the prey is strong. The result implies that the global bifurcating branches of steady state solutions are bounded loops.     

Analysis of stability and Hopf bifurcation for HIV-1 dynamics with PI and three intracellular delays

Mathematical model for the effects of protease inhibitor on the dynamics of HIV-1 infection model with three delays is proposed and analyzed. Some analytical results on the global stability of viral free steady state and infected steady state are obtained. The stability/instability of the positive steady state and associated Hopf bifurcation are investigated by analyzing the characteristic equations.     

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.     

On the Existence and Stability of a Global Subsonic Flow in a 3D Infinitely Long Cylindrical Nozzle

This paper is concerned with the problem on the global existence and stability of a subsonic flow in an infinitely long cylindrical nozzle for the 3D steady potential flow equation. Such a problem was indicated by Courant-Friedrichs in [8, p. 377]： A flow through a duct should be considered as a cal symmetry and should be determined steady, isentropic, irrotational flow with cylindriby solving the 3D potential flow equations with appropriate boundary conditions. By introducing some suitably weighted HSlder spaces and establishing a priori estimates, the authors prove the global existence and stability of a subsonic potential flow in a 3D nozzle when the state of subsonic flow at negative infinity is given.     

Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas

The existence of global smooth solutions to the multi-dimensional hydrodynamic model for plasmas of electrons and positively charged ions is shown under the assumption that the initial densities are close to a constant. The model consists of the conservation laws for the particle densities and the current densities, coupled to the Poisson equation for the electrostatic potential. Furthermore, it is proved that the particle densities converge exponentially fast to the (constant) steady state. The proof uses a higher-order energy method inspired from extended thermodynamics.     

Analytical and numerical method of finite bodies for calculation of cylindrical orthotropic shell with rectangular hole

To solve the boundary-value problem for cylindrical orthotropic shell with sizeable rectangular hole we suggest analytical and numerical method of finite bodies. For determination of the stress state of orthotropic thin-walled cylinder we use a systemof equations that exactly satisfies the equilibrium equations of orthotropic cylindrical shell. Representation of the solutions is divided into basic and self-equilibrium state. For some loads of a shell we build the basic stress state. We obtain a countable number of resolving functions that exactly satisfy the equations of a shell and describe the self-equilibrium stress state. We develop the algorithm of the analytical and numerical solutions of boundary-value problem based on approximation of the stress state of a shell by finite sum of resolving functions and propose a universal way of reduction of all conditions of the contact parts of the enclosure and the boundary conditions to minimize the generalized quadratic forms. We establish criteria under which the construction of approximate solutions coincides with the exact one.