Nonlinear Internal Damping of Wave Equations with Variable Coefficients

For wave equations with variable coefficients on regions which are not necessarily smooth, we study the energy decay rate when a nonlinear damping is applied on a suitable subrigion.     

Energy Decay for the Cauchy Problem of the Linear Wave Equation of Variable Coefficients with Dissipation

Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.     

波方程的边界镇定性的分析(英文)

利用乘子法研究了带有变系数的波方程的反馈镇定问题．文章中变系数ｑ（ｘ）分为两种情况：（Ｈ_１）ｑ和（Ｈ_２）ｑ．同时还引入了敏感反馈系数和波方程能量的指数衰减域．更为重要的是，本文统一和改善了一些以前已有的结果．     

Strichartz Estimates for Wave Equations with Coefficients of Sobolev Regularity

Wave packet techniques provide an effective method for proving Strichartz estimates on solutions to wave equations whose coefficients are not smooth. We use such methods to show that the existing results for C ^{1, 1} and C ^{1, α} coefficients can be improved when the coefficients of the wave operator lie in a Sobolev space of sufficiently high order.     

Boundary stabilization of wave equations with variable coefficients

The aim of this paper is to obtain the exponential energy decay of the solution of the wave equation with variable coefficients under suitable linear boundary feedback. Multiplier method and Riemannian geometry method are used.     

Boundary stabilization of a microbeam model

In this paper, we study the boundary stabilization of the deflection of a clamped-free microbeam, which is modeled by a sixth-order hyperbolic equation. We design a boundary feedback control, simpler than the one designed in Vatankhah et al,² that forces the energy associated to the deflection to decay exponentially to zero as the time goes to infinity. The rate in which the energy exponentially decays is explicitly given.     

Global Exact Boundary Controllability for Cubic Semi-linear Wave Equations and Klein-Gordon Equations

The authors prove the global exact boundary controllability for the cubic semi-linear wave equation in three space dimensions, subject to Dirichlet, Neumann, or any other kind of boundary controls which result in the well-posedness of the corresponding initial-boundary value problem. The exponential decay of energy is first established for the cubic semi-linear wave equation with some boundary condition by the multiplier method, which reduces the global exact boundary controllability problem to a local one. The proof is carried out in line with [2, 15]. Then a constructive method that has been developed in [13] is used to study the local problem. Especially when the region is star-complemented, it is obtained that the control function only need to be applied on a relatively open subset of the boundary. For the cubic Klein-Gordon equation, similar results of the global exact boundary controllability are proved by such an idea.     

On Local Nonreflecting Boundary Conditions for Time Dependent Wave Propagation

The simulation of wave phenomena in unbounded domains generally requires an artificial boundary to truncate the unbounded exterior and limit the computation to a finite region. At the artificial boundary a boundary condition is then needed, which allows the propagating waves to exit the computational domain without spurious reflection. In 1977, Engquist and Majda proposed the first hierarchy of absorbing boundary conditions, which allows a systematic reduction of spurious reflection without moving the artificial boundary farther away from the scatterer. Their pioneering work, which initiated an entire research area, is reviewed here from a modern perspective. Recent developments such as high-order local conditions and their extension to multiple scattering are also presented. Finally, the accuracy of high-order local conditions is demonstrated through numerical experiments.     

A Kind of Discrete Non-Reflecting Boundary Conditions for Varieties of Wave Equations

Abstract In this paper, a new kind of discrete non-reflecting boundary conditions is developed.It can be usedfor a variety of wave equations such as the acoustic wave equation, the isotropic and anisotropic elastic waveequations and the equations for wave propagation in multi-phase media and so on.In this kind of boundaryconditions,the composition of all artifical reflected waves,but not the individual reflected ones,is consideredand eliminated.Thus, it has a uniform formula for different wave equations.The velocity C_A of the composedreflected wave is determined in the way to make the reflection coefficients minimal,the value of which depends onequations.In this psper,the construction of the boundary conditions illustrated and C_A is found,numericalresults are presented to illustrate the effectiveness of the boundary conditions.     

Energy Decay for the Cauchy Problem of the Linear Wave Equation of Variable Coeffcients with Dissipation

Decay of the energy for the Cauchy problem of the wave equation of variable coeffcients with a dissipation is considered.It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coeffcients.In particular,some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.     

Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback

The aim of this paper is to investigate the uniform stabilization of Euler-Bernoulli plate equation with variable coefficients in the principle part subject to nonlinear boundary feedback laws. The exponential or rational energy decay rate is obtained by the multiplier method and the Riemannian geometry method.     

Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L₂ and L^p+1 norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.     

Periodic Solutions of Nonlinear Wave Equations with Dissipative Boundary Conditions

Applying Nash-Moser's implicit function theorem, the author proves the existence of periodic solution to nonlinear wave equation u_{tt} - u_{xx} + εg(t, x, u, u_t, u_x, u_{tt}, u_{tx}, u_{xx}) = 0 with a dissipative boundary condition, provided ε is sufficiently small.     

波动方程的非线性、非局部边值问题

用 Galerkin方法证明了波动方程的一类非线性非局部边值问题的解的存在性定理 .     

变系数分布DCNNS的全局稳定性研究

基于全局Lipschitz连续激励函数方法探讨了带有时滞的变系数Hopfield神经网络模型,借助不动点和Lyapunov泛函数确保给定的神经网络的全局渐进稳定.     

小波伽辽金方法应用于变系数波动方程

我们考虑问题K（x）uxx=ua.0＜X〈1，t≥0，其中K（x）≥a≥0，u（0，t）=g，ix（0，t）=0．这是一个不适当的方程，因为当解存在时在边界g上一个小的扰动将对它的解造成很大的改变．我们考虑存在解u（x，·）∈L^2（R）用小波伽辽金方法和Meyer多分辨分析去滤掉高频部分，从而在尺度空间Vj上得到适定的近似解．我们也可以得到问题的准确解与它在Vj上的正交投影之间的误差估计．     

带耗散项的浅水波方程的整体边界稳定性

在[0,1]区间上研究带耗散项的浅水波方程由边界反馈引起的整体指数稳定性.在控制边界条件u(0)=u_x(1)=u_(xx)(0)=δu_(xxx)(1)-εu(1)u_(xx)(1)-(?)(u(1))=0之下,证明了解的H1整体指数稳定性和H4整体渐进稳定性.     

化波动方程的非齐次边界为齐次边界

在一维波动方程初边值问题中,通过构造辅助函数的方法,分类将各种非齐次边界条件转化为齐次边界条件.     

复杂边界条件下任意变系数微分方程的半解析方法

利用有限元和矩阵函数的有关理论与变系数等效参数相结合,提出了一种求解变系数微分方程的解法,利用它能方便地处理由于各种原因导致其控制方程是变系数的微分方程,最后通过两个算例,取得了满意的结果.     

Oscillation of Neutral Equations withVariable Coefficients

51iIntroductionConsidertheneutralequationwithvariablecoefficientsInrecentyears,thedscillatotytheoryforneutralequathaiswelldeveloped.Inparticular,manydifferentsuffitientconditionsfortheoscillationofallsolutionsofEq.(l)areobtainedin[l-7j.Allthepapersmentionedexcept[6],however,alwaysassumeP(t)isinoneofthefOllowingranges:OSP(t)SlorP(t)2l.Ouraiminthispapertoeatablishnewsufficientcon-ditionsfortheoscillationofallsolutiOnsofEq.(l)underlessrestrictivehypothesesonP(t).Letm=max{Y,d}.Byasolutionof…