Stationary patterns created by cross-diffusion for the competitor-competitor-mutualist model

In this paper we consider a competitor-competitor-mutualist model with cross-diffusion. We prove some existence and non-existence results concerning non-constant positive steady-states (patterns). In particular, we demonstrate that the cross-diffusion can create patterns when the corresponding model without cross-diffusion fails.     

一类带扩散的捕食模型非常数正解的存在性

本文考虑了一类带扩散的捕食模型的平衡态问题.首先给出了正解的先验估计,进而,分别借助于能量方法和拓扑度理论得出了因参数的变化而引起的非常数正解的不存在性和存在性结果.     

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.     

一类时滞种群模型的周期正解

利用一些分析技巧和重合度理论,研究了一类时滞种群模型：Ｎ’（ｔ）＝Ｎ（ｔ）「ｂ（ｔ）－∞／∑／ｊ＝１ａｊｌｎＮ（ｔ＝ｒｊ）」周期正解的存在性,得到了一些新的判据,同时也改进了一些相关文献的结果。     

EXISTENCE OF GLOBAL SMOOTH SOLUTION TO INITIAL BOUNDARY VALUE PROBLEM FOR A VISCOELASTIC MODEL WITH RELAXATION

This paper is concerned with the initial boundary value problem for a vis-coelastic model with relaxation. Under the only assumption that the C^0-norm of theinitial data is small, without smallness hypothesis for the C^1-norm, the existence of theglobal smooth solution to the corresponding initial boundary value problem is proved.The analysis is based on some a priori estimates obtained by the “maximum principle” offirst-order quasilinear hyperbolic system.     

Non-constant positive steady states of a prey-predator system with cross-diffusions

In this paper, we study a strongly coupled elliptic system arising from a Lotka-Volterra prey-predator system, where cross-diffusions are included in such a way that the prey runs away from the predator and the predator moves away from a large group of preys. We establish the existence and non-existence of its non-constant positive solutions. Our results show that if m₁b<a<2m₁b/(1−m₁m₂) when 0<m₁m₂<1 or a>m₁b when m₁m₂?1, , d₂>0, d₃?0 and , then there exists (d₁,d₂,d₃,d₄) such that the stationary problem admits non-constant positive solutions. Otherwise, the stationary problem has no non-constant positive solution. In particular, the results indicate that its non-constant positive solutions are mainly created by the cross-diffusion d₄.     

A Lotka-Volterra symbiotic model with cross-diffusion

The main goal of this paper is to study the existence and non-existence of coexistence states for a Lotka-Volterra symbiotic model with cross-diffusion. We use mainly bifurcation methods and a priori bounds to give sufficient conditions in terms of the data of the problem for the existence of positive solutions. We also analyze the profiles of the positive solutions when the cross-diffusion parameter goes to infinity.     

Pattern formation of a coupled two-cell Brusselator model

In this paper, we study the stationary problems for the coupled two-cell Brusselator model as follows

     

中立型对数种群模型的正周期解存在性

本文利用k-集压缩算子的拓朴度抽象连续定理和某些分析技巧,研究了一类中立型对数种群模型dN/dt=N(t)[r(t)-a(t)lnN(t-σ)-b(t)d/dtlnN(t-r)]的正周期解存在性问题。得到了正周期解存在性的新结果。     

正相协下风险度量VaR样本分位数估计的渐近性质

本文研究了正相协严平稳样本下,风险度量VaR样本分位数估计的问题.利用其指数不等式和协方差不等式,获得了风险度量VaR的样本分位数估计的相合性和渐近正态性,并给出Bahadur表示.     

具有脉冲和时滞的Logistic模型的持续生存和正周期解的全局吸引性(英文)

本文研究了具有脉冲和时滞效应的Logistic模型.利用脉冲微分方程的比较定理,BohlBrower不动点定理和Lyapunov函数法,获得了系统持续生存,正周期解存在、唯一以及全局吸引的充分条件.结果表明正周期解的全局吸引性与时滞有关.     

Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials

We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under control. Our pointwise estimates decay polynomially in the velocity variable. We also show that if the initial data satisfies a Gaussian upper bound, this bound is propagated for all positive times.     

bounds in some non-standard nonlinear problems in partial differential equations

A priori bounds are determined for certain energy expressions for a class of semi-linear parabolic and hyperbolic initial-boundary value problems when a combination of the values of the solution initially and at a later time is prescribed.     

一类抛物型趋化模型的解的存在性

本文研究了一类抛物型趋化模型的解的存在性问题.利用算子半群理论,Sobolev嵌入定理及不等式技巧对解进行一些重要的先验估计,然后构造压缩映射证明了模型存在局部解.进一步构造迭代估计来说明不可能存在爆破,从而证明全局解的存在.     

A note on the regularity criteria for the Navier-Stokes equations

We consider the regularity problem for 3D Navier-Stokes equations in a bounded domain with smooth boundary. A new sufficient condition which guarantees the regularity of weak solutions on the quotient ∇p/(1+|u|^δ₁+|∇u|^δ₂) for the Navier-Stokes equations is established.     

A proof of Hessian Sobolev inequality

In this paper, taking the Hessian Sobolev inequality (0<p≤k) (X.-J. Wang, 1994 [2]) as the starting point, we give a proof of the Hessian Sobolev inequality when k<p≤k^∗, where k^∗ is the critical Sobolev embedding index of k-Hessian type. We also prove that k^∗ is optimal by one-dimensional Hardy’s inequality.     

Pattern formation in the Brusselator system

In the paper, we deal with a reaction-diffusion system well known as the Brusselator model and some improved results for the steady states of this model are presented. We first give an a priori estimates (positive upper and lower bounds) of positive steady states. Then, we obtain the non-existence and existence of positive non-constant steady states as the parameters λ, θ and b are varied, which means some certain conditions under which the pattern formation occurs or not.     

中立型对数种群模型正周期解的存在性新结果

运用k-集压缩算子的拓扑度抽象连续定理,研究了如下一类高阶微分方程的周期解存在性问题x(m)(t)= r(t)-a(t)x(t-σ)-b(t)x(m)(t-τ).并将其应用于研究一类中立型对数种群模型dN/dt= N(t) [r(t)-a(t) In N(t-σ-b(t)d/(dt)In N(t-τ)]的正周期解存在性...     

STEADY-STATE SOLUTIONS FOR A ONE-DIMENSIONAL NONISENTROPIC HYDRODYNAMIC MODEL WITH NON-CONSTANT LATTICE TEMPERATURE

A one-dimensional stationary nonisentropic hydrodynamic model for semiconductor devices with non-constant lattice temperature is studied. This model consists of the equations for the electron density, the electron current density and electron temperature, coupled with the Poisson equation of the electrostatic potential in a bounded interval supplemented with proper boundary conditions. The existence and uniqueness of a strong subsonic steady-state solution with positive particle density and positive temperature is established. The proof is based on the fixed-point arguments, the Stampacchia truncation methods, and the basic energy estimates.     

A priori estimates of the solution for the Dirichlet problem

We study the Dirichlet problem defined by u = f in and u =g in when is the half-space or the unitary rectangle, obtainingan a priori estimate of the solution. Furthermore, in both casesa concrete numerical estimation is arrived at. First we getthe a priori estimate in the case of the half-space. The problemis resolved for the rectangle by initially translating it intothe half-space and using the results we had obtained for it,so that the problem can then be reduced back to the rectangle.Because of this we first establish an extension of the Sobolevspace of second order in the rectangle to the one defined inthe half-space.