Exponential Decay Of Solutions Research Articles

Estimates of the Exponential Decay of Solutions to Linear Systems of Neutral Type with Periodic Coefficients

We consider the class of linear systems of delay differential equations with periodic coefficients. Using a special class of Lyapunov–Krasovskiĭ functionals, we establish conditions for the exponential stability of the zero solution and obtain estimates characterizing the exponential decay rate of solutions at infinity.

Global existence and exponential decay of strong solutions for the 3D incompressible MHD equations with density-dependent viscosity coefficient

This paper deals with the 3D incompressible MHD equations with density-dependent viscosity in bounded domain. The global well-posedness of strong solutions is established in the vacuum cases, provided the assumption that $${\bar{\rho }}+\Vert H_0\Vert _{L^3}$$ is suitably small with large velocity, which extends the recent work (Song in Z Angew Math Phys 69(2):23, 2018) to the global one. Furthermore, the exponential decay of the solution is also obtained.

Decay of solutions for 2D Navier–Stokes equations posed on Lipschitz and smooth bounded and unbounded domains

Initial–boundary value problems for 2D Navier–Stokes equations posed on bounded and unbounded rectangles as well as on bounded and unbounded smooth domains were considered. The existence and uniqueness of regular global solutions in bounded rectangles and bounded smooth domains as well as exponential decay of solutions on bounded and unbounded domains were established.

Energy decay and global solutions for a damped free boundary fluid–elastic structure interface model with variable coefficients in elasticity

ABSTRACT We study a free boundary fluid-structure interaction model. In the model, a viscous incompressible fluid interacts with an elastic body via the common boundary. The motion of the fluid is governed by Navier–Stokes equations while the displacement of the elastic structure is described by variable coefficient wave equations. The dissipation is placed on the common boundary between the fluid and the elastic body. Given small initial data, the global existence of the solutions of this system is proved and the exponential decay of solutions is obtained.

General decay of solutions to a viscoelastic wave equation with linear damping, nonlinear damping and source term

ABSTRACTThis paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399].

Exponential decay for waves with indefinite memory dissipation

In this work, we deal with the following wave equation with localized dissipation given by a memory term $$ u_{tt} -u_{xx} + \partial_x \Big\{ a(x)\int_{0}^{t} g(t-s)u_{x}(x,s)ds \Big\}=0. $$ We consider that this dissipation is indefinite due to sign changes of the coefficient $a$ or by sign changes of the memory kernel $g$. The exponential decay of solutions is proved when the average of coefficient $a$ is positive and the memory kernel $g$ is small.

On a general nonlinear problem with distributed delays

The paper considers a general system of ordinary differential equations appearing in the neural network theory. The activation functions are assumed to be continuous and bounded by power type functions of the states and distributed delay terms. These activation functions are not necessarily Lipschitz continuous as it is commonly assumed in the literature. We obtain sufficient conditions for exponential decay of solutions.

Existence and Exponential Decay of Solutions for a Wave Equation with Integral Nonlocal Boundary Conditions of Memory Type

ABSTRACTIn this paper, we consider a wave equation with integral nonlocal boundary conditions of memory type. First, we establish two local existence theorems by using Faedo–Galerkin method and standard arguments of density. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.

Existence and exponential decay estimates for an N-dimensional nonlinear wave equation with a nonlocal boundary condition

Motivated by the recent known results as regards the existence and exponential decay of solutions for wave equations, this paper is devoted to the study of an N-dimensional nonlinear wave equation with a nonlocal boundary condition. We first state two local existence theorems. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. The main tools are the Faedo-Galerkin method and the Lyapunov method.

On the Cauchy problem of a three-component Camassa--Holm equations

The present paper is mainly concerned with the well-posedness, blow-up phenomena and exponential decay of solution.The well-posedness for a three-component Camassa--Holm equation is established in a critical Besov space. Comparing with the result of Hu, ect. in the paper [25], a new wave-breaking solution is obtained. The exponential decay of solution in our paper covers and extents the corresponding results in [12,24,31].

On the Exponential Decay of Solutions of a Coupled System of Dissipative Benjamin–Bona–Mahony Type Equations in Domain with Moving Boundaries

We consider the Cauchy-problem in a bounded domain with moving boundaries for the nonlinear coupled dissipative system of Benjamin---Bona---Mahony type. By means of a change of variables we reduce the problem in a cylindrical domain and study the existence and uniqueness of global solutions and prove that the total energy associated with the system decays exponentially. We combine Faedo---Galerkin's method with arguments of compactness to study the existence of global solutions and energy estimates and multipliers techniques to obtain the exponential decay.

THE NON LINEAR TRANSMISSION PROBLEM WITH TIME DEPENDENT COEFFICIENTS

En este trabajo, consideramos el problema de transmisión no lineal para la ecuación de onda con coeficientes dependientes del tiempo y un damping lineal interno. Probamos la existencia global y el decaimiento exponencial de la solución. Los resultados son obtenidos por la consideración de funcionales tipo Lyapunov y un adecuado teorema de continuación única para la ecuación de onda.

The exponential decay of solutions and traveling wave solutions for a modified Camassa–Holm equation with cubic nonlinearity

The present paper is devoted to the study of persistence properties, infinite propagation, and the traveling wave solutions for a modified Camassa–Holm equation with cubic nonlinearity. We first show that persistence properties of the solution to the equation provided the initial datum is exponential decay and the initial potential satisfies a certain sign condition. Next, we get the infinite propagation if the initial datum satisfies certain compact conditions, while the solution to Eq. (1.1) instantly loses compactly supported, the solution has exponential decay as |x| goes to infinity. Finally, we prove Eq. (1.1) has a family traveling wave solutions.

On well-posedness and small data global existence for an interface damped free boundary fluid–structure model

We address a fluid–structure system which consists of the incompressible Navier–Stokes equations and a damped linear wave equation defined on two dynamic domains. The equations are coupled through transmission boundary conditions and additional boundary stabilization effects imposed on the free moving interface separating the two domains. Given sufficiently small initial data, we prove the global-in-time existence of solutions by establishing a key energy inequality which in addition provides exponential decay of solutions.

Global solution and exponential decay for a nonlinear hyperbolic equation in Banach spaces

In this work we study the existence, uniqueness and asymptotic behavior, as t→∞, of solutions of the initial value problem {u″(t)+M(t,‖u(t)‖Wβ)Au(t)+F(u(t))+(1+α‖u(t)‖Vβ)Au′(t)=0in H,u(0)=u0,u′(0)=u1, where M is a function satisfying suitable conditions, A and F are operators, V and H are two Hilbert spaces, W is a Banach space, α>0, β≥2 are two real numbers and u0, u1 are initial data. The global solutions is obtained by use of Faedo–Galerkin’s method together with a characterization of the derivative of the nonlinear term M(t,‖u(t)‖Wβ) and the Arzelà–Ascoli theorem. The exponential decay of solutions is analyzed by means of the perturbed energy method.

Spatial decay for several phase-field models

In this paper, we study the spatial behavior of three phase-field models. First, we consider the Cahn-Hilliard equation and we obtain the exponential decay of solutions under suitable assumptions on the data. Then, for the classical isothermal phase-field equation (i.e., the Allen-Cahn equation), we prove the nonexistence and the fast decay of solutions and, for the nonisothermal case governed by the Fourier law, we obtain a Phragm en-Lindel of alternative of exponential type, respectively.

Stabilization of a viscoelastic Timoshenko beam

A viscoelastic Timoshenko beam is investigated. We prove an exponential decay of solutions for a large class of kernels with weaker conditions than the existing ones in the literature. This will allow the use of other types of viscoelastic material for Timoshenko type beams than the usually used ones.

Global Existence and Exponential Decay of Solutions for a Class of System of Nonlinear Higher-Order Wave Equations with Strong Damping

Global Existence and Exponential Decay of Solutions for a Class of System of Nonlinear Higher-Order Wave Equations with Strong Damping

Global Existence and Uniform Decay of Solutions for a Coupled System of Nonlinear Viscoelastic Wave Equations with Not Necessarily Differentiable Relaxation Functions

In this paper, we consider the initial boundary value problem for a coupled system of nonlinear viscoelastic wave equations with source terms. By using the Faedo–Galerkin method, potential well theory and perturbed energy technique, we establish the global existence and exponential decay of solutions under weaker conditions on the relaxation functions that are not necessarily differentiable.

On the Exponential Decay of Solutions for Some Kirchhoff-Type Modelling Equations with Strong Dissipation

This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equations with strong dissipative and source terms in a bounded domain, where and are constants. We obtain the global existence of solutions by constructing a stable set in and show the energy exponential decay estimate by applying a lemma of V. Komornik.