EXISTENCE AND UNIQUENESS OF THE SOLUTION FOR NONLINEAR THIRD-ORDER ROBIN BOUNDARY VALUE PROBLEM WITH A TURNING POINT

The paper aims to obtain existence and uniqueness of the solution as well as asymptotic estimate of the solution for singularly perturbed nonlinear third- order Robin boundary value problem with a turning point.In order to achieve this aim,existence and uniqueness of the solution for third-order nonlinear Robin boundary value problem is derived first based on the upper and lower solutions method under relatively weaker conditions.In this manner,the goal of this paper is gained by applying the existence and uniqueness results mentioned above.     

The high-order SPDEs driven by multi-parameter fractional noises

The existence and uniqueness of the solutions are proved for a class of fourth-order stochastic heat equations driven by multi-parameter fractional noises. Furthermore the regularity of the solutions is studied for the stochastic equations and the existence of the density of the law of the solution is obtained.     

Global Classical Solutions of the Cauchy Problems for Nonlinear Vorticity Equations and Its Applications

In this paper, the definition of a vortieity equation on a compact Riemannian manifold without boundary is given. Some accurate a priori estimates are obtained with the use of the peculiarity of the equation. Then, the existence and “uniqueness” of global classical solution of the Cauchy problem for the equation is proved. As its application, the existence and “uniqueness” of Cauchy problem for barotropic nondivergent model is obtained. The model is a fundamental one in atmospheric dynamics.     

Existence and uniqueness of positive solutions of semilinear elliptic equations

This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.     

QUASI-REVERSIBLE MEASURES OF NEAREST NEIGHBOUR SPEED FUNCTIONS

Let S be a countable set with a graph structure. The process with state space \[X = {\{ 0,1\} ^s}\] is described in terms of a collection of nonnegative speed functions \[c(u, \cdot ),u \in S\]. In this paper we introduce the concept of qnasi-reversible measure for speed functions, and discuss some properties contained in the existence and uniqueness of quasi-reversible measures for the nearest neighbour speed functions, with the idea of field theory by Hou and chen[3]； In section 2, we show that qnasi-reyerisible measures are Markov random fields. A necessary and sufficient condition for the existence of quasi-reversible measures is presented. In seotion 3, a uniqueness theorem of quasi-reversible measures is given. The problem to determine the quasi-reversible measures in accordance with the speed function is discussed , for some particular cases, the quasi-reversible measures can be computed explicitly. In seotion 4, we show that if the speed functions are uniformly bounded, and each point of S has uniformly bounded boundary then the quasi-reyersible measures of the speed functions are reversible measures of the spin-flip process with the speed fnncfaons. Thus we obtain, the necessary and sufficient conditions for the existence and uniqueness of reversible measures for spin-flip process with nearest neighbour speed functions. Particularly if speed functions are defined by the nearest neighbour potential, then quasi-reversible measure exigt，thus our results can be applied to solve the uniqueness problem of Gibbs states with the nearest neighbour potential.     

BOUNDARY ELEMENT APPROXIMATION OF THE SEMI-DISCRETE PARABOLIC VARIATIONAL INEQUALITIES OF THE SECOND KIND

The boundary element approximation of the parabolic variational inequalities of the second kind is discussed. First, the parabolic variational inequalities of the second kind can be reduced to an elliptic variational inequality by using semidiscretization and implicit method in time; then the existence and uniqueness for the solution of nonlinear non-differentiable mixed variational inequality is discussed. Its corresponding mixed boundary variational inequality and the existence and uniqueness of its solution are yielded. This provides the theoretical basis for using boundary element method to solve the mixed vuriational inequality.     

WEAK AND SMOOTH GLOBAL SOLUTION FOR LANDAU-LIFSHITZ-BLOCH-MAXWELL EQUATION

This paper is devoted to investigate the existence and uniqueness of the solution of Landau-Lifshitz-Bloch-Maxwell equation. The Landau-LifshitzBloch-Maxwell equation, which fits well for a wide range of temperature, is used to study the dynamics of magnetization vector in a ferromagnetic body.If the initial data is in(H~1, L~2, L~2), the existence of the global weak solution is established. If the initial data is in(H~(m+1), H~m, H~m)(m ≥ 1), the existence and uniqueness of the global smooth solution are established.     

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS

In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.     

GLOBAL EXISTENCE OF SOLUTIONS OF CAUCHY PROBLEM FOR GENERALIZED BBM-BURGERS EQUATION

In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation （1） are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.     

Boundary value problems of discrete generalized Emden-Fowler equation

By using the critical point theory, some sufficient conditions for the existence of the solutions to the boundary value problems of a discrete generalized Emden-Fowler equation are obtained. In a special case, a sharp condition is obtained for the existence of the boundary value problems of the above equation. For a linear case, by the discrete variational theory, a necessary and sufficient condition for the existence, uniqueness and multiplicity of the solutions is also established.     

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.     

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.     

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.     

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.     

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.     

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.