On the dynamics of equations with infinite delay

We consider a system of ordinary differential equations with infinite delay. We study large time dynamics in the phase space of functions with an exponentially decaying weight. The existence of an exponential attractor is proved under the abstract assumption that the right-hand side is Lipschitz continuous. The dimension of the attractor is explicitly estimated. Research supported by the project LC06052 of the Czech Ministry of Education.     

A stability and numerical study of the solutions of a Timoshenko system with distributed delay

In this work, we analyze the stability of the semigroup associated with a Timoshenko beam model with distributed delay in the rotation angle equation. We show that the type of stability resulting from the semigroup is directly related to some model coefficients, which constitute the velocities of the system's component equations. In the case of stability of the polynomial type, we prove that rate obtained is optimal. We conclude the work performing a numerical study of the solutions and their energies, associated to discrete system.     

About partial boundary dissipation to Timoshenko system with delay

We consider the Timoshenko model with partial dissipative boundary condition with delay, and we prove that the solution decays exponentially to zero, provided the wave speed are equal; this improve earlier result due to Bassam et al and Muñoz Rivera and Naso. Moreover, consider the exponential stability to the corresponding semilinear problems.     

The finite dimensional behaviour for the higher-order nonlinear Ginzburg-Landau system inn spatial dimensions

In this paper, we study the 2m-order nonlinear Ginzburg-Landau system inn spatial dimensions. We show the existence and uniqueness of the global generalized solution, and the existence of the global attractor for this system, and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor. This project is supported by the National Natural Science Foundation of China (No. 19571010).     

Finite dimensional global attractor for a class of doubly nonlinear parabolic equations

Our aim in this paper is to study the long time behavior of a class of doubly nonlinear parabolic equations. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.     

A numerical algorithm for the nonlinear Timoshenko beam system

An initial boundary value problem is considered for the dynamic beam system Its solution is found by means of an algorithm, the constituent parts of which are the finite element method, the implicit symmetric difference scheme used to approximate the solution with respect to the spatial and time variables, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. Errors of three parts of the algorithm are estimated and, as a result, its total error estimate is obtained. A numerical example is solved.     

Energy decay to Timoshenko system with indefinite damping

We consider the classical Timoshenko system for vibrations of thin rods. The system has an indefinite damping mechanism, ie, it has a damping function a=a(x) possibly changing sign, present only in the equation for the vertical displacement. We shall prove that exponential stability depends on conditions regarding of the indefinite damping function a and a nice relationship between the coefficient of the system. Finally, we give some numerical result to verify our analytical results.     

On a class of PDEs with nonlinear distributed in space and time state‐dependent delay terms

A new class of nonlinear partial differential equations with distributed in space and time state‐dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state‐dependent delay and discuss advantages of this class. Local and long‐time asymptotic properties, including the existence of global attractor and a principle of linearized stability, are studied. Copyright © 2008 John Wiley & Sons, Ltd.     

Global dynamics of delay differential equations

In this survey paper the delay differential equation (t) = −μx(t) + g(x(t − 1)) is considered with μ ≥ 0 and a smooth real function g satisfying g(0) = 0. It is shown that the dynamics generated by this simple-looking equation can be very rich. The dynamics is completely understood only for a small class of nonlinearities. Open problems are formulated. Supported in part by the Hungarian NFSR, Grant No. T049516.     

Riesz basis property of the generalized eigenvector system of a Timoshenko beam

The Riesz basis property of the generalized eigenvector system of a Timoshenko beam with boundary feedback controls appliedto two ends is studied in this paper. The spectral property of the operator A determined by the closed loop system is investigated.It is shown that operator A has compact resolvent and generatesa C₀ semigroup, and its spectrum consistsof two branches and has two asymptotes under some conditions.Furthermore it is proved that the sequence of all generalizedeigenvectors of the system principal operator forms a Rieszbasis for the state Hilbert space.     

Uniform stabilization of a nonlinear structural acoustic model with a Timoshenko beam interface

This paper is concerned with a nonlinear model which describes the interaction of sound and elastic waves in a two‐dimensional acoustic chamber in which one flat ‘wall’, the interface, is flexible. The composite dynamics of the structural acoustic model is described by the linearized equations for a gas defined on the interior of the chamber and the nonlinear Timoshenko beam equations on the interface. Uniform stability of the energy associated with the interactive system of partial differential equations is achieved by incorporating a nonlinear feedback boundary damping scheme in the equations for the gas and the beam. Copyright © 2006 John Wiley & Sons, Ltd.     

On the stabilization of the Timoshenko system by a weak nonlinear dissipation

In this paper we consider the following Timoshenko‐type system: Without imposing any restrictive growth assumption on g at the origin, we establish a general decay result depending on g and α. Copyright © 2008 John Wiley & Sons, Ltd.     

A note on the finite fractal dimension of the global attractors for dissipative nonlinear Schrödinger‐type equations

In this article, we present, throughout two basic models of damped nonlinear Schrödinger (NLS)–type equations, a new idea to bound from above the fractal dimension of the global attractors for NLS‐type equations. This could answer the following open issue: consider, for instance, the classical one‐dimensional cubic nonlinear Schrödinger equation $$u_t+i{u}_{xx}+i|u{|}^{2}u+\gamma u=f,f\in {\mathrm{??}}^2(?).$$ “How can we bound the fractal dimension of the associate global attractor without the need to assume that the external forcing term f has some decay at infinity (that is belonging to some weighted Lebesgue space)?”     

A stability result of a Timoshenko system with a delay term in the internal feedback

In this paper, we consider a Timoshenko system with a delay term in the feedback and prove a stability result. The beam is clamped at the endpoints and has, in addition to an internal damping, a feedback with a delay.Under an appropriate assumption on the weights of the two feedbacks, we prove the well-posedness of the system and establish an exponential decay result for the case of equal-speed wave propagation.     

非线性时滞差分方程的全局吸引性

得到了具有多重时滞非线性差分方程x(n 1)-x(n) ∑i=1^k pi(n)fi(x(gi(n)))=0,n=0,1,2,…的每个解趋于零的充分条件。     

广义Ginzburg—Landau方程组的有限维行为

本文得到了广义Ｇｉｎｚｂｕｒｇ－Ｌａｎｄａｕ方程组的解的整体存在性和唯一性，同时得到了有限维整体吸引子的存在性。     

带有热记忆的非均匀柔性结构的长时间动力行为

本文研究了带有热效应的非均匀柔性结构方程,并且该热效应符合Coleman-Gurtin定律.利用半群方法,建立了系统的整体适定性.主要结论是该系统的长时间动力行为.本文证明了系统的拟稳定性,整体吸引子的存在性以及整体吸引子具有有限的分形维数.此外,还证明了指数吸引子的存在性.     

Dimension of the global attractor for damped nonlinear wave equations

An estimate on the Hausdorff dimension of the global attractor for damped nonlinear wave equations, in two cases of nonlinear damping and linear damping, with Dirichlet boundary condition is obtained. The gained Hausdorff dimension is bounded and is independent of the concrete form of nonlinear damping term. In the case of linear damping, the gained Hausdorff dimension remains small for large damping, which conforms to the physical intuition.