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.     

Lack of exponential decay of a coupled system of wave equations with memory

In this work, we consider a coupled system of wave equations with memory only acting in one of the equations of the system. We show that the solution of this system has a polynomial rate of decay as time tends to infinity, but does not have exponential decay.     

Stability of solution for a mixture of thermoelastic of type III

In this work, we analyze the existence, uniqueness, and asymptotic behavior of solution to the model of a thermoelastic mixture of type III. We establish sufficient conditions to guarantee the exponential decay of solutions. When the decay is not of exponential type, we prove that the solutions decay polynomially and we find the optimal polynomial decay rate. Copyright © 2017 John Wiley & Sons, Ltd.     

A boundary condition with memory for the Kirchhoff plate equations with non‐linear dissipation

This paper is concerned with the existence of solutions for the Kirchhoff plate equation with a memory condition at the boundary. We show the exponential decay to the solution, provided the relaxation function also decays exponentially. Copyright © 2005 John Wiley & Sons, Ltd.     

Vibrations of a beam with a damping tip body

We investigate a mathematical model for the dynamics of a beam with a tip body that experiences damping. The damping is due to granular material which partially fills the tip body. We establish the existence of the unique solution to the model and analyze the model. Among other things, we establish exponential energy decay when damping is present.     

Finite-parameter feedback stabilization of original Burgers' equations and Burgers' equation with nonlocal nonlinearities

We study the problem of global exponential stabilization of original Burgers' equations and the Burgers' equation with nonlocal nonlinearities by controllers depending on finitely many parameters. We investigate both equations by employing controllers based on finitely many Fourier modes and the latter equation by employing finitely many volume elements. To ensure global exponential stabilization, we have provided sufficient conditions on the control parameters for each problem. We also show that solutions of the controlled equations are steering a concrete solution of the non-controlled system as t → ∞ with an exponential decay rate.     

Polynomial stability of a coupled system of wave equations weakly dissipative

In this work, we consider a coupled system of wave equation. We show that the solution of this system has a polynomial rate of decay as time tends to infinity, but does not have exponential decay. We presented a class of examples of application of the main result.     

On the exponential stability of solutions of periodic systems of the neutral type with several delays

We obtain conditions for the exponential stability of the zero solution of linear periodic systems of differential equations of the neutral type with several constant delays, which are stated in terms of a Lyapunov–Krasovskii functional of a special form. We derive estimates that specify the decay rate of solutions at infinity.     

大初值的抛物型守恒律方程解的大时间性态(英文)

In this paper,we study the large-time behavior of periodic solutions for parabolic conservation laws.There is no smallness assumption on the initial data.We firstly get the local existence of the solution by the iterative scheme,then we get the exponential decay estimates for the solution by energy method and maximum principle,and obtain the global solution in the same time.     

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.     

Decay solution for the renewal equation with diffusion

In this paper we consider age structured equation with diffusion under nonlocal boundary condition and nonnegative initial data. We prove existence, uniqueness and the positivity of the solution to the above problem. Our main result is to get an exponential decay of the solution for large times toward such a study state. To this end we prove a weighted Poincaré–Wirtinger’s type inequality in unbounded domain.     

A boundary condition with memory in elasticity

In this paper, we study the stability of solutions of the n-dimensional nonhomogeneous and anisotropic elastic system with memory condition working at the boundary. We show that such dissipation is strong enough to produce exponential decay to the solution, provided the relaxation function also decays exponentially.     

Asymptotic Stability of Traveling Wave Solutions for Perturbations with Algebraic Decay

For a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the stability of traveling wave solutions is investigated. If the initial perturbation of the traveling wave profile decays at an algebraic rate, then the solution is shown to converge to a shifted wave profile at a corresponding temporal algebraic rate, and optimal intermediate results that combine temporal and spatial decay are obtained. The proofs are based on a general interpolation principle which says that algebraic decay results of this form always follow if exponential temporal decay holds for perturbation with exponential spatial decay and the wave profile is stable for general perturbations.     

About Asymptotic Behavior for a Transmission Problem in Hyperbolic Thermoelasticity

We consider a transmission problem in thermoelasticity with memory. We show the exponential decay of the solution in case of radially symmetric situations, as time goes to infinity.      

On the Kirchhoff plates equations with thermal effects and memory boundary conditions

In this paper we study the existence of weak and strong global solutions and uniform decay of the energy to the Kirchhoff plates equations with thermal effect and memory conditions working at the boundary. We show that the dissipation produced by the memory effect not depend on the present values of temperature gradient. That is, we show that the dissipation produced by memory effect is strong enough to produce exponential decay of the solution provided the relaxation functions also decays exponentially. When the relaxation functions decays polynomially, we show that the solution decays polynomially with the same rate.     

On the time polynomial decay in elastic solids with voids

In this paper we investigate the temporal decay behavior of the solutions of the one-dimensional problem in various theories of continua with voids. It has been proved that the coupling of the elastic structure with porous microstructure is weak in the sense that in many situations the temporal decay of solutions is slow. We have considered some theories of porous continua when the deformation-rate tensor or time-rate or porosity function or thermal effects is present. We have proved that the decay cannot be controlled by a negative exponential. The natural question now is whether there exist or not a polynomial rate of decay of the solution in some appropriate norms. In this paper we consider some cases where the decay is slow and we obtain polynomial decay estimates. In concrete we consider the case when only the viscoelastic effect is present, the case when the motion of voids is assumed to be quasi-static and the porous viscosity is present and we finish with the case of the porous-elasticity when thermal effect is coupled.     

Exponential decay bounds for nonlinear heat problems with Robin boundary conditions

We investigate the behavior of the solution of a nonlinear heat problem, when Robin conditions are prescribed on the boundary ∂Ω × (t > 0), Ω a bounded R ² domain. We determine conditions on the geometry and data sufficient to preclude the blow up of the solution and to obtain an exponential decay bound for the solution and its gradient.     

THE EXPONENTIAL STABILIZATION FOR A SEMILINEAR WAVE EQUATION WITH LOCALLY DISTRIBUTED FEEDBACK

This paper considers the exponential decay of the solution to a damped semilinear wave equation with variable coefficients in the principal part by Riemannian multiplier method. A differential geometric condition that ensures the exponential decay is obtained.     

On decay of solutions to nonlinear Schrödinger equations

We present general results on exponential decay of finite energy solutions to stationary nonlinear Schrödinger equations. Under certain natural assumptions we show that any such solution is continuous and vanishes at infinity. This allows us to interpret the solution as a finite multiplicity eigenfunction of a certain linear Schrödinger operator and, hence, apply well-known results on the decay of eigenfunctions.

     

Exponential decay bounds for nonlinear heat problems with Robin boundary conditions

We investigate the behavior of the solution of a nonlinear heat problem, when Robin conditions are prescribed on the boundary ∂Ω × (t > 0), Ω a bounded R ² domain. We determine conditions on the geometry and data sufficient to preclude the blow up of the solution and to obtain an exponential decay bound for the solution and its gradient. Supported by the University of Cagliari.