The degenerate hyperbolic problem for the two-dimensional nonlinear wave system

This paper constructs a local classical solution of degenerate hyperbolic problem for the two-dimensional nonlinear wave system. To deal with the parabolic degeneracy, we introduce a new set of coordinates to transform the nonlinear wave system to a new system that has explicitly singularity-regularity structures. By constructing a weighted metric space, we establish the existence of solution for the new system. Returning the solution to the original variables, we obtain the existence of the classical solution for the nonlinear wave system with degenerate boundary value problem.     

AN EXPONENTIAL STABILITY PROBLEM ON THE SMALL DELAY LINEAR SYSTEM

1IntroductionLetXbea'Banachspace,andB(X)betheBanachspaceofcontinuouslinearoperatorsfromXintoX.LetT(t)bea(CO)selltigroup,andletAbeitsinfinitesilllalgenerator.WedenotethedomainandtherangeofAbyD(A)andR(A).LetBbelongtoB(X).FOrthelinearsystem:andthedelaylinearsystem:wherer>0,((.)EC([--.,o],x)={fif:[--r,oj~X,fiscontinuous},r(t)EC[o,.)'05r(t)5r.Weconsidertileexponentialstabilityequivalencebetweellthesolutionof(1)and(2),whichllleans:ifthereexistM,a>0,sllththatIIS(f)II5Me--"'(f20),whereS(…     

An efficient numerical method for singular perturbation problems

In this paper, we propose a method for the numerical solution of singularly perturbed two-point boundary-value problems (BVPs). First, we develop two schemes to integrate initial–value problem (IVP) for system of two first-order differential equations, and then by using these schemes we solve the BVP. Precisely, we convert the second-order BVP into a system of first-order differential equations, and then apply the numerical schemes to obtain the solution. In order to get an initial condition for the system, we use the asymptotic approximate solution. Error estimates are derived and numerical examples are provided to illustrate the present method.     

A semigroup approach to the Gnedenko system with single vacation of a repairman

We investigate the Gnedenko system with one repairman who can take vacations. Our main focus is on the time asymptotic behaviour of the system. Using C ₀-semigroup theory for linear operators we first prove the well-posedness of the system and the existence of a unique positive dynamic solution given an initial value. Then by analysing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator we show that the dynamic solution converges strongly to the steady state solution. Thus we obtain asymptotic stability of the dynamic solution.     

Mathematical and physical aspect of the solution of the Cauchy problem for a system of equations describing a nonlocal model of propagation of heat with finite speed

In our paper we present a new system of equations describing a nonlocal model of propagation of heat with finite speed in three-dimensional space. Such a system of equations is described by a system of integral – differential equations. At first using the modiffied Cagniard de Hoop method, we construct the fundamental solution of this system of equations. On the basis of the constructed fundamental solution we obtain the explicite formulate of the solution of the Cauchy problem for this system of equations and applying the method of Sobolev and Biesov spaces, we get L^p – L^q time decay estimate for the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

具有热储备的可修复平行系统在由常规错误引起失效下的渐进稳定性

讨论了具有热储备和两个独立相同部件的平行系统在由常规错误引起失效下的渐进稳定性.首先,利用Banach空间的Volttera算子方程得到了非负动态解的存在唯一性;然后,利用强连续线性算子半群理论证明了系统正的动态解的存在唯一性,而由于初始值不在定义域内,故得到的是mild解.但在t＞0时系统古典解存在唯一,所以此时mild解即为古典解.最后,利用线性算子半群稳定性的结果,证明了该动态解在范数意义下收敛到稳态解,进而得到了系统的渐进稳定性.     

The system of generalized vector equilibrium problems with applications

In this paper, we introduce the system of generalized vector equilibrium problems which includes as special cases the system of generalized implicit vector variational inequality problems, the system of generalized vector variational and variational-like inequality problems and the system of vector equilibrium problems. By using a maximal element theorem, we establish existence results for a solution of these systems. As an application, we derive existence results for a solution of a more general Nash equilibrium problem for vector-valued functions.     

Systems of generalized variational inequalities and their applications ∗

In this paper, we first introduce the system of generalized implicit variational inequalities and prove the existence of its solution. Then we derive existence results for systems of generalized variational and variational like inequalities and system of variational inequalities. As applications, we establish some existence results for a solution to the system of optimization problems which includes the Nash equilibrium problem as a special case     

The System of Vector Quasi-Equilibrium Problems with Applications

In this paper, we consider the system of vector quasi-equilibrium problems with or without involving -condensing maps and prove the existence of its solution. Consequently, we get existence results for a solution to the system of vector quasi-variational-like inequalities. We also prove the equivalence between the system of vector quasi-variational-like inequalities and the Debreu type equilibrium problem for vector-valued functions. As an application, we derive some existence results for a solution to the Debreu type equilibrium problem for vector-valued functions.     

Local existence of the solution to the initial‐boundary value problem in nonlinear thermomicroelasticity theory

We prove a theorem about local existence (in time) of the solution to the first initial‐boundary value problem for a nonlinear system of equation of the thermomicroelasticity theory. At first, we prove existence, uniqueness and regularity of the solution to this problem for the associated linearized system by using the method of semi‐group theory. Next, basing on this theorem, we prove an energy estimate for the solution to the linearized system by applying the method of Sobolev space. At the end, using the Banach fixed point theorem, we prove that the solution of our nonlinear problem exists and is unique. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类病毒与抗体的反应扩散方程组解的性质

对一类病毒与抗体的反应扩散方程组利用变量变换的方法得到与其具有同解性的反应扩散方程.在一定假设条件下,研究此方程解的一些性质,再由紧性得到满足原假设的解的收敛序列,从而得到此方程解的存在性、惟一性与收敛性.借助于方程与方程组的同解性,最终得到了反应扩散方程组解的性质.     

Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

In this paper we deal with the problem of existence of a smooth solution of the Hamilton–Jacobi–Bellman–Isaacs (HJBI for short) system of equations associated with nonzero-sum stochastic differential games. We consider the problem in unbounded domains either in the case of continuous generators or for discontinuous ones. In each case we show the existence of a smooth solution of the system. As a consequence, we show that the game has smooth Nash payoffs which are given by means of the solution of the HJBI system and the stochastic process which governs the dynamic of the controlled system.     

一类奇异拟线性椭圆型方程组的可解性

研究了一类P-Laplacian方程组边值问题正径向整体解的存在性和唯一性.首先,利用隐函数定理证明了该问题的局部解的存在性与唯一性,以及解对初值的连续依赖性.最后,证明了该问题存在唯一的正径向整体解.     

二维浅水波方程的间断周期解和间断孤立波解讨论

本文讨论地球流体运动浅水波模式中间断周期解与间断孤立波解.在系统的非平衡点即奇点附近考虑轨线性质时,我们发现只要引入广义解的概念(分片光滑连续解),就会产生间断周期解并得到了间断周期解的条件.当系统在退化的过程中,发现系统此时会产生间断的孤立波解,与此同时其它物理量也产生了间断.这里我们发现,一般认为在超高速情况下解会产生间断,然而在非超高速时也会产生间断现象.本文讨论了上述一系列问题得到了间断解的解析解表达式,并把这一事实与飑线的实例进行比较,得到了不少类似之处.     

Uniform persistence and periodic solutions for a discrete predator-prey system with delays

In this paper, we deal with a discrete predator-prey system with delay. We first give a sufficient condition for the uniform persistence of the system. Assuming that the coefficients in the system are periodic, by generalizing the Yoshizawa's theorem on the existence of periodic solution for ordinary differential equations to the difference equations with delays, we obtain the existence of a periodic solution basing on the uniform persistence result.     

The limiting absorption principle for the two-dimensional inhomogeneous anisotropic elasticity system

In this work we establish the limiting absorption principle for the two-dimensional steady-state elasticity system in an inhomogeneous aniso- tropic medium. We then use the limiting absorption principle to prove the existence of a radiation solution to the exterior Dirichlet or Neumann boundary value problems for such a system. In order to define the radiation solution, we need to impose certain appropriate radiation conditions at infinity. It should be remarked that even though in this paper we assume that the medium is homogeneous outside of a large domain, it still preserves anisotropy. Thus the classical Kupradze's radiation conditions for the isotropic system are not suitable in our problem and new radiation conditions are required. The uniqueness of the radiation solution plays a key role in establishing the limiting absorption principle. To prove the uniqueness of the radiation solution, we make use of the unique continuation property, which was recently obtained by the authors. The study of this work is motivated by related inverse problems in the anisotropic elasticity system. The existence and uniqueness of the radiation solution are fundamental questions in the investigation of inverse problems.

     

Analysis and finite element approximation of an optimal control problem for the Oseen viscoelastic fluid flow

In this article we study a boundary control problem for an Oseen-type model of viscoelastic fluid flow. The existence of a unique optimal solution is proved and an optimality system is derived by the first-order necessary condition. We investigate finite element approximations to a solution of the optimality system, and a solution algorithm for the system based on the gradient method.     

由软件和硬件构成的串联可修计算机系统的适定性及渐近稳定性

讨论由软件和硬件构成的串联可修计算机系统的时间依赖解,运用C_0-半群理论及算子理论,证明系统的适定性和时间依赖非负解的存在唯一性.通过研究系统相应算子的谱特征,得到系统时间依赖解的渐近稳定性.     

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L^∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.     

具有状态反馈脉冲控制的种群互惠动力系统的研究

研究具有状态反馈脉冲控制的种群互惠动力系统.首先利用微分方程几何理论和后继函数的方法得到一般系统阶1周期解的存在条件;然后研究了一类特殊系统,说明了该系统在一定条件下存在唯一的阶1周期解,并且给出了该阶1周期解轨道渐近稳定的条什,此外还探讨了该系统阶2周期解的存在性问题.