Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application

We establish the uniqueness of ground states of some coupled nonlinear Schrödinger systems in the whole space. We firstly use Schwartz symmetrization to obtain the existence of ground states for a more general case. To prove the uniqueness of ground states, we use the radial symmetry of the ground states to transform the systems into an ordinary differential system, and then we use the integral forms of the system. More interestingly, as an application of our uniqueness results, we derive a sharp vector-valued Gagliardo-Nirenberg inequality.     

On the Semiconductor System

In this paper, the semiconductor system is discussed. The existence and uniqueness or the global solution or the carrier transport problem are obtained. Under the condition that the width in some direction of the domain being sufficiently small, the existence and uniqueness of the solution of the steady states are proved. It is also proved that the solution of the carrier transport problem tends to the solution or the steady states problem exponentially when t goes to infinity.     

带有扩散和Beddington-DeAngelis响应函数的捕食模型的正平衡态

讨论了带有扩散和Beddington-DeAngelis响应函数的捕食模型在齐次Dirichlet边界条件下的正平衡态,给出了正平衡态存在的一个充要条件,当猎物种群相互干扰强的时候,得到正平衡态的稳定性与唯一性。     

COEXISTENCE STATES OF A STRONGLY COUPLED PREY-PREDATOR MODEL

This paper is concerned with a strongly coupled prey-predator model with homogeneous Dirichlet boundary conditions. The existence and uniqueness of coexistence states are discussed.     

Analysis of diffusive two-competing-prey and one-predator systems with Beddington-Deangelis functional response

In this paper, a diffusive two-competing-prey and one-predator system with Beddington-DeAngelis functional response is considered. The sufficient and necessary conditions for the existence of coexistence states are provided using the fixed point index theory developed. In addition, the stability and uniqueness of coexistence states are investigated. Finally, this paper discusses the sufficient conditions for extinction and permanence of the time-dependent system.     

Quantitative properties on the steady states to a Schrödinger–Poisson–Slater System

In this paper, we obtain the existence, uniqueness and asymptotic behavior of steady states to a class of Schrödinger-Poisson-Slater System.     

Steady states of radially symmetric diffusion processes

This note is concerned with the steady states of some diffusion processes arising from some physical problems. We are interested in the discussion of radially symmetric steady states, which lead to a degenerate and singular equation. Under some natural structural condition, we obtain the existence and uniqueness of nonnegative solutions.     

Positive Steady States of a Prey-predator Model with Diffusion and Non-monotone Conversion Rate

In this paper, we study the positive steady states of a prey-predator model with diffusion throughout and a non-monotone conversion rate under the homogeneous Dirichlet boundary condition. We obtain some results of the existence and non-existence of positive steady states. The stability and uniqueness of positive steady states are also discussed.     

Existence and uniqueness of coexistence states for an elliptic system coupled in the linear part

In this paper, we discuss the existence, uniqueness and stability of coexistence states for an elliptic system coupled in the linear part. The linear part of this system is co-operative. Our approaches are based on the maximum principle and some properties of linear cooperative system.     

An elliptic system modeling two subpopulations

In this paper we study an elliptic system modeling two subpopulations of the same species competing for resources. We show the existence and uniqueness of coexistence states of the system by giving necessary and sufficient conditions.     

Additive functionals of semi-Markov processes with absorption

Given a semi-Markov process with an arbitrary set of states, a criterion is obtained for the attainability of a certain isolated subset of states and for finiteness of the average attainment time. An equation is given for the average of an additive functional of a process with absorption, existence and uniqueness conditions are deduced for the solution of that equation in a given class of functions, and an integral representation is obtained for the solution.     

A competition un-stirred chemostat model with virus in an aquatic system

A reaction–diffusion system of two bacteria species competing a single limiting nutrient with the consideration of virus infection is derived and analysed. Firstly, the well-posedness of the system, the existence of the trivial and semi-trivial steady states, and some prior estimations of the steady states are given. Secondly, a single species subsystem with virus is studied. The stability of the trivial and semi-trivial steady states and the uniform persistence of the subsystem are obtained. Further, taking the infective ability of virus as a bifurcation parameter, the global structure of the positive steady states and the effect of virus on the positive steady states are established via bifurcation theory and limiting arguments. It shows that the backward bifurcation may occur. Some sufficient conditions for the existence, uniqueness and stability of the positive steady state are also obtained. Finally, some sufficient conditions on the existence of the positive steady states for the full system are derived by using the fixed point index theory. Some results on persistence or extinction for the full system are also obtained.     

Well-posedness of a degenernate parabolic equation issuing from two-dimensional perfect fluid dynamics

A degenerate parabolic equation of convection-diffusion type has been proposed by Robert and Sommeria in [12] to describe the relaxation towards statistical equilibrium states in 2D incompressible perfect fluid dynamics. The paper is concerned with the Cauchy problem for this equation. The local existence of a variational soluation is obtained in using the decrease of the (negative) mixing entropy and Schauder theorem. A smoothing effect is used when proving the uniqueness of the variational solutions by youdovitch's method. Finally, global existence of solutions and their asymptotic convergence towards Gibbs states are shown for a large class of initial data.     

KMS states on C*-algebras associated to expansive maps

Using Walters' version of the Ruelle-Perron-Frobenius Theorem we show the existence and uniqueness of KMS states for a certain one-parameter group of automorphisms on a C*-algebra associated to a positively expansive map on a compact metric space.

     

Coexistence states for a diffusive one-prey and two-predators model with B–D functional response

In this paper, we study a diffusive one-prey and two-predators system with Beddington–DeAngelis functional response. The sufficient and necessary conditions for the existence of coexistence states are obtained by means of the fixed point index theory. In addition, the stability and uniqueness of coexistence states are investigated. Finally, we give the sufficient conditions for extinction and permanence of the time-dependent system.     

一类微生物连续培养竞争系统的定性分析

研究一类微生物连续培养竞争系统的解的结构,分析了平衡点的稳定性及平衡点附近极限环存在唯一性,证明了该系统存在正向不变集.     

Steady states in plasma physics—the Vlasov–Fokker–Planck equation

In this paper we investigate the non-linear Vlasov–Fokker–Planck (VFP) equation, a both physically and mathematically interesting modification of Vlasov's equation, which describes a plasma in a thermal bath. We prove existence, uniqueness and representation results for steady states of the VFP equation both in the case of a mollified interaction potential and for the VFP–Poisson system. The uniqueness and representation results are of special interest since they distinguish special solutions of the Vlasov equation.     

Initial-boundary value problems for anisotropic Landau-Lifshitz equations in three or four space dimensions

In this paper, we will show the existence of partially regular solutions to the initial-boundary value problem for Landau-Lifshitz equations with nonpositive anisotropy constants in three or four space dimensions. The partial regularity is proved up to the boundary both for the Dirichlet problem and for the Neumann problem. In addition, for the Neumann case, a generalized stability condition which ensures the partial regularity is given. For equations with positive or negative anisotropy coefficients, we will give two results of existence and uniqueness for the solutions corresponding to ground states.     

Passivity and complementarity

This paper studies the interaction between the notions of passivity of systems theory and complementarity of mathematical programming in the context of complementarity systems. These systems consist of a dynamical system (given in the form of state space representation) and complementarity relations. We study existence, uniqueness, and nature of solutions for this system class under a passivity assumption on the dynamical part. A complete characterization of the initial states and the inputs for which a solution exists is given. These initial states are called consistent states. For the inconsistent states, we introduce a solution concept in the framework of distributions.     

Dynamics of a simple food chain model with a ratio-dependent functional response

This paper considers a diffusive ratio-dependent simple food chain model. The sufficient conditions for the existence and non-existence of coexistence states are provided. In addition, this paper investigates the uniqueness of coexistence states and examines the global attractor of the time-dependent system. Moreover, in view of extinction results, the domino effect and biological control are discussed.