General decay for a quasilinear system of viscoelastic equations with nonlinear damping

In this paper, we consider a system of coupled quasilinear viscoelastic equations 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.     

Energy decay rate of the thermoelastic Bresse system

In this paper, we study the energy decay rate for the thermoelastic Bresse system which describes the motion of a linear planar, shearable thermoelastic beam. If the longitudinal motion and heat transfer are neglected, this model reduces to the well-known thermoelastic Timoshenko beam equations. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. Actually, the corresponding energy decays exponentially like the classical one-dimensional thermoelastic system. However, the third wave equation about the vertical displacement is only weakly damped. Thus the decay rate of the energy of the overall system is still unknown. We will show that the exponentially decay rate is preserved when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement. Otherwise, only a polynomial type decay rate can be obtained. These results are proved by verifying the frequency domain conditions.      

Comparison of decay of solutions to two compressible approximations to Navier-Stokes equations

In this article, we use the decay character of initial data to compare the energy decay rates of solutions to different compressible approximations to the Navier- Stokes equations.We show that the system having a nonlinear damping term has slower decay than its counterpart with an advection-like term. Moreover, me characterize a set of initial data for which the decay of the first system is driven by the difference between the full solution and the solution to the linear part, while for the second system the linear part provides the decay rate.     

Stabilization by memory effects: Kirchhoff plate versus Euler–Bernoulli plate

In this paper we study the asymptotic behavior of solutions of a dissipative coupled system where we have interactions between a Kirchhoff plate and an Euler–Bernoulli plate. The dissipative mechanism is given by memory terms that act either collaboratively (in both equations) or unilaterally (in only one equation). We show that the solutions of this system decay to zero sometimes exponentially and other times polynomially. We found explicit decay rates that depend on the fractional exponents of the memory in each of the following cases: when the memory only acts in the Kirchhoff equation, or only in the Euler–Bernoulli equation, or in both. We also show that all decay rates found are the best. The results obtained are surprising for the following facts: in the collaborative case, the best decay rates of the system are given by the worst decay rates of the uncoupled equations, and in the unilateral case, we conclude that the memory effects in the Euler–Bernoulli equation dissipate the system more slowly than memory effects in the Kirchhoff equation.     

Radially symmetric solutions of the p-Laplacian in perforated-like domain with nonlocal boundary condition

In this paper, we study the stability of solutions to a von Kármán system for Kirchhoff plate equations with a memory condition working at the boundary. We show that such dissipation is strong enough to produce exponential decay of the solution provided the relaxation functions also decay exponentially. When the relaxation functions decay polynomially, we show that the solution decays polynomially.     

Dynamics for Controlled 2D Generalized MHD Systems with Distributed Controls

We study the dynamics of a piecewise (in time) distributed optimal control problem for Generalized MHD equations which model velocity tracking coupled to magnetic field over time. The long-time behavior of solutions for an optimal distributed control problem associated with the Generalized MHD equations is studied. First, a quasi-optimal solution for the Generalized MHD equations is constructed; this quasi-optimal solution possesses the decay (in time) properties. Then, some preliminary estimates for the long-time behavior of all solutions of Generalized MHD equations are derived. Next, the existence of a solution of optimal control problemis proved also optimality system is derived. Finally, the long-time decay properties for the optimal solutions is established.     

Upper bounds of the rates of decay for solutions of the Boussinesq equations

In this paper,upper bounds of the L2-decay rate for the Boussinesq equations are considered.Using the L2 decay rate of solutions for the heat equation,and assuming that the solutions of the Boussinesq equations are smooth,we obtain the upper bounds of L2 decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data.The decay results may then be obtained by passing to the limit of approximating sequences of solutions.The main tool is the Fourier splitting method.     

Mathematical and physical aspect of the solution of the Cauchy problem for a system of equations describing a nonlocal model of propagation of heat with finite speed

In our paper we present a new system of equations describing a nonlocal model of propagation of heat with finite speed in three-dimensional space. Such a system of equations is described by a system of integral – differential equations. At first using the modiffied Cagniard de Hoop method, we construct the fundamental solution of this system of equations. On the basis of the constructed fundamental solution we obtain the explicite formulate of the solution of the Cauchy problem for this system of equations and applying the method of Sobolev and Biesov spaces, we get L^p – L^q time decay estimate for the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

THE IMPROVED FOURIER SPLITTING METHOD AND DECAY ESTIMATES OF THE GLOBAL SOLUTIONS OF THE CAUCHY PROBLEMS FOR NONLINEAR SYSTEMS OF FLUID DYNAMICS EQUATIONS

Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems. We will couple together the elementary uniform energy estimates of the global weak solutions and a well known Gronwall''s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980''s to study the optimal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations. As applications, the decay estimates with sharp rates of the global weak solutions of the Cauchy problems for $n$-dimensional incompressible Navier-Stokes equations, for the $n$-dimensional magnetohydrodynamics equations and for many other very interesting nonlinear evolution equations with dissipations can be established.     

Asymptotic behavior for coupled abstract evolution equations with one infinite memory

In this paper, we consider two coupled abstract linear evolution equations with one infinite memory acting on the first equation. Our work is motivated by the recent results of [42], where the authors considered the case of two wave equations with one convolution kernel converging exponentially to zero at infinity, and proved the lack of exponential decay. On the other hand, the authors of [42] proved that the solutions decay polynomially at infinity with a decay rate depending on the regularity of the initial data. Under a boundedness condition on the past history data, we prove that the stability of our abstract system holds for convolution kernels having much weaker decay rates than the exponential one. The general and precise decay estimate of solution we obtain depends on the growth of the convolution kernel at infinity, the regularity of the initial data, and the connection between the operators describing the considered equations. We also present various applications to some distributed coupled systems such as wave-wave, Petrovsky-Petrovsky, wave-Petrovsky, and elasticity-elasticity.     

Frequency domain approach for the polynomial stability of a system of partially damped wave equations

In this paper, we study the stability of a system of wave equations which are weakly coupled and partially damped. Using a frequency domain approach based on the growth of the resolvent on the imaginary axis, we establish the polynomial energy decay rate for smooth initial data. We show that the behavior of the system is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two equations.     

Stabilization of the uniform Timoshenko beam by one locally distributed feedback

In this article, we study the energy decay rate for an elastic Timoshenko system. This system consists of two coupled wave equations. Only the equation about the rotation angle is damped by one locally distributed feedback at the neighbourhood of the boundary. The equation for the transverse displacement of the beam is only indirectly damped through the coupling. First, we establish an exponential energy decay rate in the case of the same speed of propagation. Next, when the wave speeds are different, a polynomial-type decay rate is obtained. These results are proved by verifying the frequency domain conditions.     

Large Time Behaviour of Solutions to the Navier-Stokes Equations in H Spaces

In this paper we establish the decay of the homogeneous H norms for solutions to the Navier Stokes equations in two dimensions. The rates of decay are obtained by means of the Fourier splitting method. The rate obtained is optimal in the sense that it coincides with the rates for solutions to the heat system.     

The exponential decay of a system of nonlinear wave equations

The exponential decay of a system of nonlinear wave equations with initial boundary values is considered. We have some sufficient conditions that ensure that the energy admits exponential decay by a compactness uniqueness argument and the energy estimates.     

一个带有阻尼项和源项的非线性粘弹性方程组解的整体衰减

本文考虑了一个带有非线性阻尼项的粘弹性方程组.通过使用扰动能量的方法,我们得到了整体解的能量泛函依据松弛函数的衰减速率按指数衰减或者多项式衰减.     

Stability results for the approximation of weakly coupled wave equations

In this Note, we consider the approximation of two coupled wave equations with internal damping. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay (since the spectrum of the spatial operator associated with the undamped system satisfies the generalized gap condition).     

Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms

In this paper, we consider a system of two coupled wave equations with dispersive and viscosity dissipative terms under Dirichlet boundary conditions. The global existence of weak solutions as well as uniform decay rates (exponential one) of the solution energy are established.     

Numerical analysis of nonlinear model of excited carrier decay

This paper presents a mathematical model for photo-excited carrier decay in a semiconductor. Due to the carrier trapping states and recombination centers in the bandgap, the carrier decay process is defined by the system of nonlinear differential equations. The system of nonlinear ordinary differential equations is approximated by linearized backward Euler scheme. Some a priori estimates of the discrete solution are obtained and the convergence of the linearized backward Euler method is proved. The identifiability analysis of the parameters of deep centers is performed and the fitting of the model to experimental data is done by using the genetic optimization algorithm. Results of numerical experiments are presented.     

Polynomial stability of solutions for a system of non-linear viscoelastic equations

In this article, we consider an interacting system consisting of two weakly-coupled viscoelastic equations. This work proves a uniform stabilization result without assuming any decay rate of the relaxation function provided that the kernels have some flat zones, the size of the flat zones are sufficiently small.     

EXISTENCE AND DECAY OF WEAK SOLUTIONS TO DEGENERATE DAVEY-STEWARTSON EQUATIONS

In this paper, the authors consider a system of degenerate Davey-Stewartson equations. They prove the global existence of weak solutions in some weighted function spaces and the decay of weak solutions in some anisotropic spaces for appropriate initial data.