scholarly journals Exponential and Polynomial Decay for a Laminated Beam with Fourier's Type Heat Conduction

2019 ◽  
Author(s):  
Wenjun Liu ◽  
Weifan Zhao
Keyword(s):  
Heat Conduction ◽  
Energy Method ◽  
Rotation Angle ◽  
Polynomial Decay ◽  
Structural Damping ◽  
Laminated Beam ◽  
Wave Speeds ◽  
One Dimensional ◽  
Exponentially Stable ◽  
Well Posed

In this paper, we study the well-posedness and asymptotics of a one-dimensional thermoelastic laminated beam system either with or without structural damping, where the heat conduction is given by Fourier's law effective in the rotation angle displacements. We show that the system is well-posed by using Lumer-Philips theorem, and prove that the system is exponentially stable if and only if the wave speeds are equal, by using the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem. Furthermore, we show that the system with structural damping is polynomially stable provided that the wave speeds are not equal, by using the second-order energy method.

scholarly journals Exponential and Polynomial Decay for a Laminated Beam with Fourier's Type Heat Conduction

2017 ◽  
Author(s):  
Wenjun Liu ◽  
Weifan Zhao
Keyword(s):  
Heat Conduction ◽  
Rotation Angle ◽  
Polynomial Decay ◽  
Well Posedness ◽  
Laminated Beam ◽  
Wave Speeds ◽  
One Dimensional ◽  
Beam System ◽  
Exponentially Stable ◽  
Well Posed

In this paper, we study the well-posedness and the asymptotic behavior of a one-dimensional laminated beam system, where the heat conduction is given by Fourier's law effective in the rotation angle displacements. We show that the system is well-posed by using the Hille-Yosida theorem and prove that the system is exponentially stable if and only if the wave speeds are equal. Furthermore, we show that the system is polynomially stable provided that the wave speeds are not equal.

Asymptotic stability for a laminated beam with structural damping and infinite memory

2020 ◽  
Vol 25 (10) ◽  
pp. 1979-2004 ◽  
Author(s):  
Wenjun Liu ◽  
Xiangyu Kong ◽  
Gang Li
Keyword(s):  
Wave Propagation ◽  
Relaxation Function ◽  
Rotation Angle ◽  
Structural Damping ◽  
Laminated Beam ◽  
Infinite Memory ◽  
One Dimensional ◽  
Special Cases ◽  
Lack Of Exponential Stability ◽  
Well Posed

In this paper, we consider a one-dimensional laminated beam with structural damping and an infinite memory acting on the effective rotation angle. Under appropriate assumptions imposed on the relaxation function, we show that the system is well-posed by using the Hille–Yosida theorem, and then we establish general decay results, from which exponential and polynomial decays are only special cases, in the case of equal speeds of wave propagation as well as that of nonequal speeds. In the particular case when the wave propagation speeds are different and the relaxation function decays exponentially, we show the lack of exponential stability.

scholarly journals Lack of Exponential Decay for a Laminated Beam with Structural Damping and Second Sound

2017 ◽  
Author(s):  
Wenjun Liu ◽  
Xiangyu Kong ◽  
Gang Li
Keyword(s):  
Heat Conduction ◽  
Exponential Decay ◽  
Polynomial Decay ◽  
Second Sound ◽  
Structural Damping ◽  
Stability Number ◽  
Laminated Beam ◽  
One Dimensional ◽  
The Stability ◽  
Math Phys

In previous work (Z. Angew. Math. Phys. 68(2), 2017), Apalara considered a one dimensional thermoelastic laminated beam under Cattaneo’s law of heat conduction and proved the exponential and polynomial decay results depend on the stability number χT . In this paper, we continue to study the same system and show that the solution of the concerned system lacks of exponential decay result in the case χT ≠ 0 which solves the open problem proposed by Apalara (Z. Angew. Math. Phys. 68(2), 2017).

scholarly journals Well-posedness and exponential decay for a laminated beam in thermoelasticity of type III with delay term

Mathematica ◽  
2021 ◽  
Vol 63 (86) (1) ◽  
pp. 58-76
Author(s):  
Douib Madani ◽  
Salah Zitouni ◽  
Djebabla Abdelhak
Keyword(s):  
Asymptotic Behaviour ◽  
Exponential Decay ◽  
Well Posedness ◽  
Laminated Beam ◽  
Wave Speeds ◽  
Delay Term ◽  
Type Iii ◽  
Asymptotic Behaviour Of Solutions ◽  
Exponentially Stable ◽  
Well Posed

We study the well-posedness and asymptotic behaviour of solutions to a laminated beam in thermoelasticity of type III with delay term in the first equation. We show that the system is well-posed by using Lumer-Philips theorem and prove that the system is exponentially stable if and only if the wave speeds are equal.

scholarly journals On the Stabilization for Laminated Beam with Infinite Memory and Different Speeds of Wave Propagation

2017 ◽  
Author(s):  
Gang Li ◽  
Xiangyu Kong
Keyword(s):  
Wave Propagation ◽  
Differential Equations ◽  
Boundary Control ◽  
Rotation Angle ◽  
General Decay ◽  
Laminated Beam ◽  
Wave Speeds ◽  
Infinite Memory ◽  
One Dimensional ◽  
Order Energy

In this work, we consider a one-dimensional laminated beam in the case of non-equal wave speeds with only one infinite memory on the effective rotation angle. In this case, we establish the general decay result for the energy of solution without any kind of internal or boundary control. The main result is obtained by applying the method used in Guesmia et al. (Electron. J. Differential Equations 193: 1-45, 2012) and the second-order energy.

scholarly journals Uniform stability for a type III thermoelastic laminated beam with structural damping

2022 ◽  
Author(s):  
Fayssal Djellali
Keyword(s):  
Heat Flux ◽  
Exponential Stability ◽  
Semigroup Theory ◽  
Stability Result ◽  
Uniform Stability ◽  
Well Posedness ◽  
Structural Damping ◽  
Laminated Beam ◽  
Wave Speeds ◽  
Lack Of Exponential Stability

In this work, we consider a thermoelastic laminated beam with structural damping, where the heat flux is given by Green and Naghdi theories. We establish the well-posedness of the system using semigroup theory. Moreover, under the condition of equal wave speeds, we prove an exponential stability result for the considered system. In the case of lack of exponential stability we show that the solution decays polynomially.

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

2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Djamel Ouchenane
Keyword(s):  
Heat Conduction ◽  
Stability Result ◽  
Well Posedness ◽  
Second Sound ◽  
Timoshenko System ◽  
Wave Speeds ◽  
One Dimensional ◽  
Delay Term ◽  
Semigroup Method ◽  
Cattaneo’S Law

AbstractIn this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with a delay term in the feedback. The heat conduction is given by Cattaneo's law. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem. Furthermore, an exponential stability result is shown without the usual assumption on the wave speeds. To achieve our goals, we make use of the semigroup method and the energy method.

Heat Conduction in a Thin Film which Reflects Size and Memory Effects Induced by an Ultra-Fast Laser

2015 ◽  
Vol 723 ◽  
pp. 873-877
Author(s):  
Yu Dong Mao ◽  
Ming Tian Xu
Keyword(s):  
Thin Film ◽  
Heat Conduction ◽  
Thermal Wave ◽  
Memory Effects ◽  
Wave Effect ◽  
Conduction Model ◽  
Wave Speeds ◽  
One Dimensional ◽  
Higher Temperature ◽  
Cv Model

An improved heat conduction model which reflects size and memory effects based on the CV model is proposed to simulate a one-dimensional thermal transport problem in a thin film induced by the ultra-fast laser. The results show that a thermal wave effect is appeared in heat conduction. We find that for the improved CV model in this work a larger Knudsen number will lead to a higher temperature. Although the improved CV model and the CV model lead to the similar thermal wave behavior, the thermal wave speeds for the two models are different.

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