HOMOGENIZATION OF SOME LINEAR AND SEMILINEAR SCHRDINGER EQUATIONS WITHREAL POTENTIAL

1 IntroductionIn this paPer, we collsider the initial and boundaxy value probIem of linear and sentilillearScl1r5dinger equation as fOllows:l ie(x):5:i".,'<">"'>.'<",,'>,"=' 1:!\:?0t:)t>, (l1)wllere E > 0, fl C RN is a bounded open dolllain and T is an ar…     

Propagation of Mild Singularities in Higher Dimensional Thermoelasticity

The propagation of mild singularities for the semilinear model of three-dimensional thermoelasticity is studied. It is shown that the propagation picture of such singularities of the solution to the semilinear model coincides with one of the solutions to the corresponding linear model. As a simple consequence of our method, a similar result for the full semilinear Cauchy problem of one-dimensional thermoelasticity is also presented.     

一类边界控制半线性抛物型系统的逼近能控性

本文考虑一类Ｄｉｒｉｃｈｌｅｔ型边界控制半线性抛物型系统，应用半群理论和非线性分析的方法，证明了系统解映射的连续性，给出了系统逼近能控的一个充分条件．对于Ｎｅｕｍａｎｎ型边界控制或混合型边界控制的半线性抛物型系统，可以得到类似的结果．     

Controllability of Some Nonlinear Systems in Hilbert Spaces

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.     

Global attractor for the semilinear thermoelastic problem

In this paper we consider a class of semilinear thermoelastic problems. The global attractor for this semilinear thermoelastic problem with Dirichlet boundary condition is obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

带跳的分数倒向重随机微分方程及相应的随机积分偏微分方程

本文首次把Poisson随机测度引入分数倒向重随机微分方程,基于可料的Girsanov变换证明由Brown运动、Poisson随机测度和Hurst参数在(1/2,1)范围内的分数Brown运动共同驱动的半线性倒向重随机微分方程解的存在唯一性.在此基础上,本文定义一类半线性随机积分偏微分方程的随机黏性解,并证明该黏性解由带跳分数倒向重随机微分方程的解唯一地给出,对经典的黏性解理论作出有益的补充.     

关于一类半线性高阶障碍问题解的存在性与正则性

本文考虑一类带互补边界条件的四阶半线性椭圆型变分不等式,应用临界点理论,证明了当半线性项具有超线性增长时解的存在性,并利用L~p估计得到了解的正则性.     

Delta-waves for Semilinear Hyperbolic Cauchy Problems

This paper deals with the propagation of strong singularities for constant coefficient semilinear hyperbolic equations and systems. Limits of regularized solutions are computed as the initial data converge to derivatives of Dirac measures on lower dimensional submanifolds. A general method is given which applies whenever the fundamental solution to the principal part is an integrable measure. Particular cases are semilinear first order systems in one space variable and the semilinear Klein-Gordon equation in at most three space variables.     

Propagation of singularities for Cauchy problems of semilinear thermoelastic systems with microtemperatures

The propagation of singularities of solutions to the Cauchy problem of a semilinear thermoelastic system with microtemperatures in one space variable is studied. First, by using a diagonalization argument of phase space analysis, the coupled thermoelastic system with microtemperatures will be decoupled microlocally. Second, using a classical bootstrap argument, the property of finite propagation speed of singularities for the semilinear thermoelastic system is obtained. Finally, it is also shown that the microlocal weak singularities propagate along the null bicharacteristics of the hyperbolic operators of the coupled semilinear system (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

半线性抛物最优控制问题全离散插值系数有限元方法的收敛性分析

本文考虑全离散插值系数有限元方法求解半线性抛物最优控制问题,其中控制变量用分片常数函数逼近,状态变量和对偶状态变量用分片线性函数逼近.对于方程中的半线性项,先用插值系数技巧处理,再用牛顿迭代法求解.通过引入一些辅助变量和投影算子,并利用有限元空间的逼近性质,得到半线性抛物最优控制问题插值系数有限元方法的收敛性结果;数值算例结果验证了理论结果的正确性.     

Existence and the number of solutions of nonresonant semilinear equations and applications to boundary value problems

In this paper, we study the existence and the finiteness of the number of solutions to nonlinear equations and to nonresonant semilinear equations involving nonlinear perturbations of Fredholm maps of index zero. Applications are given to semilinear elliptic boundary value problems and to semilinear parabolic and hyperbolic periodic-boundary value problems.     

Exact Controllability of Semilinear Evolution Systems and Its Application

In this paper, we obtain several abstract results concerning the exact controllability of semilinear evolution systems. First, we prove the null local exact controllability of semilinear first-order systems by means of the contraction mapping principle; in this case, we do not assume any compactness. Next, we derive the global and/or local exact controllability of semilinear second-order systems by means of the Schauder fixed-point theorem; in this case, we assume only the embedding of the related spaces having some compactness, which is reasonable for many concrete problems. Our main result shows that the observability of the dual of the linearized system implies the exact controllability of the original semilinear system. Finally, we apply our abstract results to the exact controllability of the semilinear wave equation.     

一个椭圆特征问题解对参数的连续性

用变分方法证明了一个限制在球面上的椭圆特征问题解对参数（球面半径）的连续性，从而得到相应的不带限制的椭圆特征问题的解分枝。     

Propagation of Singularities of Solutions to Hyperbolic‐Parabolic Coupled Systems

In this paper, the authors study the propagation of singlarities for a semilinear hyperbolic‐parabolic coupled system, which comes from the model of thermoelasticity. Both of the Cauchy problem and the problem inside of a domain are considered. We obtain that the microlocal singularities of solutions to the semilinear hyperbolic‐parabolic coupled system are propagated along null bicharacteristics of the hyperbolic operator by using the theory of paradifferential operators. Furthermore, for the Cauchy problem of the semilinear coupled system, if the initial data have singularities at the origin, we prove that the solutions have the same order regularity with respect to spatial variables as in hyperbolic problems in the forward characteristic cone issuing from the origin, which improves the previous results for semilinear systems in thermoelasticity.     

kth Order convergence for semilinear elliptic boundary value problems

The method of quasilinearization coupled with the method of upper and lower solutions, is applied to the semilinear elliptic boundary value problems. At the same time, the result of kth convergence for the semilinear boundary value problems is obtained via the idea of Taylors approximation.     

Multigrid method for coupled semilinear elliptic equation

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.     

Comparison Method of Stability Analysis of Semilinear Time-Delay Stochastic Evolution Equation

Abstract

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin–Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples.     

The problem of start control for a class of semilinear distributed systems of Sobolev type

A result on the solvability of the Cauchy problem for a semilinear equation of Sobolev type in a Banach space is obtained with the help of the theory of degenerate operator semigroups. The result is used for investigating the problem of start control in the corresponding system. Abstract results are illustrated by the example of the semilinear Dzektser equation.     

A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems

We consider the finite element discretization of semilinear parabolic optimization problems subject to pointwise in time constraints on mean values of the state variable. In order to control the feasibility violation induced by the discretization, error estimates for the semilinear partial differential equation are derived. Based upon these estimates, it can be shown that any local minimizer of the semilinear parabolic optimization problems satisfying a weak second-order sufficient condition can be approximated by the discretized problem. Rates for this convergence in terms of temporal and spatial discretization mesh sizes are provided. In contrast to other results in numerical analysis of optimization problems subject to semilinear parabolic equations, the analysis can work with a weak second-order condition, requiring growth of the Lagrangian in critical directions only. The analysis can then be conducted relying solely on the resulting quadratic growth condition of the continuous problem, without the need for similar assumptions on the discrete or time semidiscrete setting.     

Banach空间中半线性发展方程的复合单调迭代技巧

本文对具有正则锥的序Banach空间中半线性发展方程，利用C0-半群理论和复合单调迭代技巧，构造出了抽象迭代格式，并建立了最大、最小(拟)解的存在性．