GENERAL DECAY OF SOLUTIONS FOR A VISCOELASTIC EQUATION WITH NONLINEAR DAMPING AND SOURCE TERMS

The initial boundary value problem for a viscoelastic equation ｜ u t ｜ ρ u tt △u-△u tt ＋ t 0 g（ts）△u（s）ds ＋ ｜ u t ｜ m u t = ｜ u ｜ p u in a bounded domain is considered, where ρ, m, p 〉 0 and g is a nonnegative and decaying function. The general uniform decay of solution energy is discussed under some conditions on the relaxation function g and the initial data by adopting the method of [14, 15, 19]. This work generalizes and improves earlier results in the literature.

GENERAL DECAY FOR A VISCOELASTIC EQUATION OF VARIABLE COEFFICIENTS WITH A TIME-VARYING DELAY IN THE BOUNDARY FEEDBACK AND ACOUSTIC BOUNDARY CONDITIONS

A variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term is considered.Under suitable assumptions, general decay results of the energy are established via suitable Lyapunov functionals and some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term and the elements of the matrix A and the kernel function g.

GENERAL DECAY OF SOLUTIONS FOR A VISCOELASTIC EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING AND NONLINEAR BOUNDARY DAMPING-SOURCE INTERACTIONS

A viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary/interior sources is considered in a bounded domain. Under appropriate assumptions ira- posed on the source and the damping, we establish uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function.

GENERAL DECAY FOR A QUASILINEAR SYSTEM OF VISCOELASTIC EQUATIONS WITH NONLINEAR DAMPING

In this paper, we consider a system of coupled quasilinear viscoelastic equa- tions with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which extends some existing results concerning a general decay for a single equation to the case of system, and a nonlinear system of viscoelastic wave equations to a quasilinear system.

GENERAL DECAY FOR A POROUS-THERMOELASTIC SYSTEM WITH MEMORY: THE CASE OF NONEQUAL SPEEDS

The aim of this paper is to establish a general decay result for a one-dimensional porous elastic system with different speeds of wave propagation in the presence of macrotem- perature effect and visco-porous dissipation.

GENERAL DECAY OF A TRANSMISSION PROBLEM FOR KIRCHHOFF TYPE WAVE EQUATIONS WITH BOUNDARY MEMORY CONDITION

In this paper, we investigate the influence of boundary dissipation on the de-cay property of solutions for a transmission problem of Kirchhoff type wave equation with boundary memory condition. By introducing suitable energy and Lyapunov functionals, we establish a general decay estimate for the energy, which depends on the behavior of relaxation function.

GENERAL DECAY FOR A DIFFERENTIAL INCLUSION OF KIRCHHOFF TYPE WITH A MEMORY CONDITION AT THE BOUNDARY

In this article, we consider a differential inclusion of Kirchhoff type with a memory condition at the boundary. We prove the asymptotic behavior of the corresponding solutions. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.

General Decay Pathwise Stability of Neutral Stochastic Differential Equations with Unbounded Delay

Without the linear growth condition, by the use of Lyapunov function, this paper estab- lishes the existence^and-uniqueness theorem of global solutions to a class of neutral stochastic differen- tim equations with unbounded delay, and examines the pathwise stability of this solution with general decay rate. As an application of our results, this paper also considers in detail a two-dimensional unbounded delay neutral stochastic differential equation with polynomial coefficients.

General Decay Synchronization for Recurrent Neural Networks with Mixed Time Delays

This paper studies the general decay synchronization(GDS)of a class of recurrent neural networks(RNNs)with general activation functions and mixed time delays.By constructing suitable Lyapunov-Krasovskii functionals and employing useful inequality techniques,some sufficient conditions on the GDS of considered RNNs are established via a type of nonlinear control.In addition,one example with numerical simulations is presented to illustrate the obtained theoretical results.

General Decay for a Thermoelastic Problem of a Microbeam with Gurtin-Pipkin Thermal Law

In this paper,we study the well-posedness and the asymptotic stability of a one-dimensional thermoelastic microbeam system,where the heat conduction is given by Gurtin-Pipkin thermal law.We first establish the well-posedness of the system by using the semigroup arguments and Lumer-Phillips theorem.We then obtain an explicit and general formula for the energy decay rates through perturbed energy method and some properties of the convex functions.

ELASTIC MEMBRANE EQUATION WITH MEMORY TERM AND NONLINEAR BOUNDARY DAMPING:GLOBAL EXISTENCE,DECAY AND BLOWUP OF THE SOLUTION

In this paper we consider the Elastic membrane equation：with memory term and nonlinear boundary damping： Under some appropriate assumptions on the relaxation function h and with certain initial data, the global existence of solutions ：and a general decay for the energy are established using the multiplier technique. Also, ＇we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a nonlinear damping.

General Energy Decay of Solutions for a Wave Equation with Nonlocal Damping and Nonlinear Boundary Damping

In this paper,we consider a nonlinear wave equation with nonlocal damping and nonlinear boundary damping.We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping by using of the multiplier technique from the idea of Martinez[1],Our result extends and improves the result in the literature such as the work by Louredo,Ferreira de Araujo and Mi-randain[2]in which only exponential energy decay is considered.Furthermore,we get also the energy decay for the equation with nonlocal damping only but without nonlinear boundary damping.

Energy Decay in Thermoelasticity with Viscoelastic Damping of General Type

In this paper we consider an n-dimensional thermoelastic system with viscoelastic damping. We establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Our result allows a larger class of relaxation functions and generalizes previous results existing in the literature.

General Energy Decay of Solutions for a Weakly Dissipative Kirchhoff Equation with Nonlinear Boundary Damping

In this article, we study the weak dissipative Kirchhoff equation under nonlinear damping on the boundary We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping. Our result extends and improves some results in the literature such as the work by Zhang and Miao （2010） in which only exponential energy decay is considered and the work by Zhang and Huang （2014） where the energy decay has been not considered.

STABILITY OF A VON KARMAN EQUATION WITH INFINITE MEMORY

In this paper, we consider a von Karman equation with infinite memory. For yon Karman equations with finite memory, there is a lot of literature concerning on existence of the solutions, decay of the energy, and existence of the attractors. However, there are few results on existence and energy decay rate of the solutions for yon Karman equations with infinite memory. The main goal of the present paper is to generalize previous results by treating infinite history instead of finite history.

Stabilization and Blow-up in an Infinite Memory Wave Equation with Logarithmic Nonlinearity and Acoustic Boundary Conditions

In this paper,the authors consider the stabilization and blow up of the wave equation with infinite memory,logarithmic nonlinearity and acoustic boundary conditions.The authors discuss the existence of global solutions for the initial energy less than the depth of the potential well and investigate the energy decay estimates by introducing a Lyapunov function.Moreover,the authors establish the finite time blow up results of solutions and give the blow up time with upper bounded initial energy.

On the General Stability of a Viscoelastic Wave Equation with an Integral Condition

We consider a nonlocal boundary value problem for a viscoelastic equation with a Bessel operator and a weighted integral condition and we prove a general decay result.We also give an example to show that our general result gives the optimal decay rate for ceratin polynomially decaying relaxation functions.This result improves some other results in the literature.

Asymptotic Behavior for a Viscoelastic Wave Equation with a Time-varying Delay Term

The following viscoelastic wave equation with a time-varying delay term in internal feedback ｜ut｜^ρutt-△u-△utt＋∫0^tg（t-s）△u（s）ds＋μ1ut（x,t）＋μ2ut（x,t-τ））=0,is considered in a bounded domain. Under appropriate conditions on μ1, μ2 and on the kernel g, we establish the general decay result for the energy by suitable Lyapunov functionals.