Global attractor for a class of nonlinear lattices

We consider a class of nonlinear lattices with nonlinear damping

(0.1)     

具结构阻尼的拟线性膜方程的吸引子及其上半连续性

本文研究具结构阻尼的拟线性膜方程utt+△2u+(-△)αut+△φ(△u)+f(u)=g的适定性以及解的长时间动力学行为,其中α∈(1,2),旨在研究耗散指标α对方程解的适定性和长时间动力学行为的影响.本文证明非线性项φ(s)存在一个依赖于耗散指标α的临界指数pα=N+4(α-1)/(N-4(α-1))(N=3,4)...     

Attractors for General Operators

In this article we introduce the notion of a minimal attractor for families of operators that do not necessarily form semigroups. We then obtain some results on the existence of the minimal attractor. We also consider the nonautonomous case. As an application, we obtain the existence of the minimal attractor for models of Cahn-Hilliard equations in deformable elastic continua.     

Global attractor for an extensible beam equation with localized nonlinear damping and linear memory

In this paper, we study the existence of global attractors for the extensible beam equation with localized nonlinear damping and linear memory. Copyright © 2011 John Wiley & Sons, Ltd.     

一类二阶非线性阻尼微分方程解的振动性质

研究了一类二阶非线性阻尼微分方程解的振动性质.在一定条件下,建立了四个新的振动性定理,推广和改进了已知的结果.     

Attractors of dissipative hyperbolic equations with singularly oscillating external forces

We study a uniform attractor $\mathcal{A}^\varepsilon $ for a dissipative wave equation in a bounded domain Ω ? ?ⁿ under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g ₀(x, t)+ ε^?α g ₁ (x, t/ε), x ∈ Ω, t ∈ ?, where α > 0, 0 < ε ≤ 1. In E = H ₀ ¹ × L ₂, this equation has an absorbing set B ^ε estimated as ‖B ^ε‖ _E ≤C ₁+C ₂ε^?α and, therefore, can increase without bound in the norm of E as ε → 0+. Under certain additional constraints on the function g ₁(x, z), x ∈ Ω, z ∈ ?, we prove that, for 0 < α ≤ α ₀, the global attractors $\mathcal{A}^\varepsilon $ of such an equation are bounded in E, i.e., $\parallel \mathcal{A}^\varepsilon \parallel _E \leqslant C_3 $ , 0 < ε ≤ 1. Along with the original equation, we consider a “limiting” wave equation with external force g ₀(x, t) that also has a global attractor $\mathcal{A}^0 $ . For the case in which g ₀(x, t) = g ₀(x) and the global attractor $\mathcal{A}^0 $ of the limiting equation is exponential, it is established that, for 0 < α ≤ α ₀, the Hausdorff distance satisfies the estimate $dist_E (\mathcal{A}^\varepsilon ,\mathcal{A}^0 ) \leqslant C\varepsilon ^{\eta (\alpha )} $ , where η(α) > 0. For η(α) and α ₀, explicit formulas are given. We also study the nonautonomous case in which g ₀ = g ₀(x, t). It is assumed that sufficient conditions are satisfied for which the “limiting” nonautonomous equation has an exponential global attractor. In this case, we obtain upper bounds for the Hausdorff distance of the attractors $\mathcal{A}^\varepsilon $ from $\mathcal{A}^0 $ , similar to those given above.     

Oscillation criteria for nonlinear second order differential equations with damping

Summary Using the integral average method, we give some new oscillation criteria for the second order differential equation with damped term (a(t)Ψ(x(t))K(x'(t)))'+p(t)K(x'(t))+q(t)f(x(t))=0, t<span style='font-size:10.0pt; font-family:"Lucida Sans Unicode"'>≧t₀. These results improve and generalize the oscillation criteria in<span lang=EN-US style='font-size:10.0pt;mso-ansi-language:EN-US'>[1], because they eliminate both the differentiability of p(t) and the sign of p(t), q(t). As a consequence, improvements of Sobol's type oscillation criteria are obtained.     

Unbounded One-Dimensional Global Attractor for the Damped Sine-Gordon Equation

Summary. The dynamical behavior of the damped sine-Gordon equation with homogeneous Neumann boundary condition is studied. It is shown that the equation has an unbounded one-dimensional global attractor in a suitable functional space when the ``damping' and the ``diffusing' are not very small. Received March 16, 1999; accepted December 10, 1999     

具有双记忆项的热弹耦合梁方程的整体吸引子

研究了具有双记忆项的非线性热弹耦合梁方程,利用已知的研究结果给出解的适定性定理,其次通过先验估计并结合常用不等式证明系统存在有界吸收集,且利用标准方法验证半群的渐近紧性,得到整体吸引子的存在性.     

二阶非线性差分方程的振动准则

建立了二阶非线性差分方程△(rn△xn-1) Pn△xn-1 qnf(xn)=0的振动准则，所得定理推广和改进了文献中的结果．     

Some new oscillation theorems for second order nonlinear elliptic equations with damping

In this paper, some new oscillation criteria are obtained for second order elliptic differential equations with damping of the form

     

Attractors for a plate equation with nonlocal nonlinearities

This paper contains results on well‐posedness, stability, and long‐time behavior of solutions to a class of plate models subject to damping and source terms given by the product of two nonlinear components [EQUATION1] where Ω is a bounded open set of R ⁿ with smooth boundary, γ ,ρ ?0 and are nonlocal functions. The main result states that the dynamical system {S (t )}_{t ?0} associated with this problem has a compact global attractor. In addition, in the limit case γ  = 0, it is also shown that {S (t )}_{t ?0} has a finite dimensional global attractor by using an approach on quasi‐stability because of Chueshov–Lasiecka (2010). Copyright © 2017 John Wiley & Sons, Ltd.     

One-dimensional global attractor for the damped and driven sine-Gordon equation

The damped and driven sine-Gordon equation with Neumann boundary conditions is studied. It is shown that it has a one-dimensional global attractor in a suitable functional space when the “damping” and the “diffusing” are not very small. Project supported by the National Natural Science Foundation of China.     

General decay of energy for a viscoelastic equation with nonlinear damping

In this paper we are concerned with a nonlinear viscoelastic equation with nonlinear damping. The general uniform decay of the energy is obtained. Copyright © 2008 John Wiley & Sons, Ltd.     

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.     

Attractors of nonlinear dynamical systems with a weakly monotone measure

This paper is devoted to the study of attractive sets for dynamical systems in a metric space with a measure. It is assumed that the measure of a set of points in the phase space is increasing along the flow. We prove that an invariant set is an attractor for almost all initial conditions under some extra assumptions. For a system of autonomous ordinary differential equations, we present attractivity conditions in terms of the divergence with a density function. Unlike previous results in the literature, our approach allows the use of a wider class of density functions if the divergence vanishes on a set of positive measure. As an example, the attitude stabilization problem of a rigid body is solved by using an affine feedback control for the kinematic equations in terms of quaternions.     

On oscillation of nonlinear second order differential equation with damping term

In this paper we investigate the oscillation of nonlinear differdntial equation with damping term. Some new oscillation criteria for the equation are obtained.     

Oscillation of even order nonlinear functional differential equations with damping

This paper is concerned with a class of even order nonlinear damped differential equations