Thermoelastic System Research Articles

Asymptotic behavior of the porous elastic system with dual phase lag model: Classical versus second spectrum perspectives

AbstractThis paper aims to analyze the energy decay of the thermoelastic porous system. The dual‐phase lag theory is used to model heat transfer. We consider two perspectives: the classical approach and the second spectrum approach. For the classical approach, the well‐posedness is obtained via the semigroup theory and the system is exponentially stable under equal wave speed conditions. On the opposite, we show a polynomial decay. On the other hand, the well‐posedness of the truncated system is obtained via the Faedo Galerkin method, and the system is exponentially stable without any assumptions on the physical parameters.

A note on asymptotic profiles for the thermoelastic plate system

We investigate the Cauchy problem for the thermoelastic plate system associated with Newton’s law of cooling, where optimal growth ( n ⩽ 4 n\leqslant 4 ) or decay ( n ⩾ 5 n\geqslant 5 ) estimates and asymptotic profiles of solutions for large-time are studied. Especially, the additional lower-order term in the temperature equation weakens decay rates of the vertical displacement, and leads to a new leading term comparing with the classical thermoelastic plates.

Exponential stability for porous thermoelastic systems with Gurtin-Pipkin flux

In this article, we study the stability of a porous thermoelastic system with Gurtin-Pipkin flux. Under suitable assumptions for the derivative of the heat flux relaxation kernel, we establish the existence and uniqueness of solution by applying the semigroup theory, and prove the exponential stability of system without considering the wave velocity by the means of estimates of the resolvent

Exponential and Polynomial Decay Rates of a Porous Elastic System with Thermal Damping

This paper concerns a linear porous thermoelastic system, where the heat conduction is given by Cattaneo’s law. By using the theory of the semigroup of the linear operator, the well-posedness is proven. By constructing some suitable multipliers, we establish exponential and polynomial decay rates of the system depending on a stability number π . In addition, we remove the assumption that the constant b is positive in the previous works, which extends some existing results in the literature.

On the decay of a porous thermoelastic system with microtemperature

On the decay of a porous thermoelastic system with microtemperature

Controllability of a thermoelastic system

In this paper we present a null controllability result for a thermoelastic Rayleigh system. Instead of working directly with the control system, we obtain the controlled system as the modulus of elasticity in shear tends to infinity in the corresponding thermoelastic Mindlin–Timoshenko system. Our results follow the seminal book of Lagnese and Lions (Rech. Math. Appl. 6(1988)) where the controllability of a Kirkhhoff model is proposed as the limit of a controlled Mindlin–Timoshenko one. We use estimates for some eigenvalues of the beam model that were obtained in (SIAM J. Control Optim. 47 (2008) 1909–1938) and the recent paper of Komornik and Tenenbaum (Evolution Equations and Control Theory 4(3) (2015) 297–314) where explicit estimates for systems with real and complex eigenvalues are proposed.

Do equal speed condition and exponential stability relate for the truncated thermoelastic Timoshenko system under Green Naghdi law?

Over the years, the stabilization Timoshenko systems with dissipative features have piqued the interest of researchers. The study of Timoshenko systems under various damping effects has resulted in a significant number of studies. When nonphysical assumptions of equal wave velocities are used in stabilization, the expected exponential decay of the energy solution is attained in all recent research. In this study, we analyze a one-dimensional thermooelastic Timoshenko type system in the setting of the second frequency spectrum, where the assumption of equal wave speed is not required for exponential decay to occur. In fact, According to Elishakoff’s studies [Elishakoff, Advances in Mathematical Modeling and Experimental Methods for Materials and Structures: The Jacob Aboudi Volume, Dordrecht, The Netherlands: Springer, pp. 249–254, 2009.], we consider the so-called truncated version of the Timoshenko system, and we added a thermoelastic damping according to Green Naghdi law of heat conduction. We first use Faedo–Galerkin approximation to verify the system’s global well-posedness. Using a Lyapunov functional we establish an exponential stability without assuming the condition of equal wave speed. A numerical scheme is introduced and analyzed. Finally, assuming extra regularity on the solution, we get some a priori error estimates and we present some numerical results which demonstrate the exponential behavior of the solution. This result significantly improves the previous results in the literature in which equal wave velocities are used to obtain exponential stability.

Pullback dynamics for a class of non-autonomous Lamé thermoelastic system

Pullback dynamics for a class of non-autonomous Lamé thermoelastic system

Stabilization of a thermoelastic diffusion system of type II damped and delayed

ABSTRACT The present paper is devoted to the study of exponential stabilization of a one-dimensional thermoelastic diffusion problem of type II for isotropic and homogeneous materials with frictional damping and time delay. The system of equations under consideration is a coupling of three hyperbolic equations involving three variables considering frictional damping and time delay on each variable. Using semigroup theory, we prove the well-posedness by the Lumer–Phillips theorem and the exponential stability by exploring the dissipative properties of the linear operator associated with the model through the Gearhart–Huang–Pruss theorem.

Stability of a linear thermoelastic Bresse system with second sound under new conditions on the coefficients

AbstractIn this paper, we discuss the stability of a linear one‐dimensional thermoelastic Bresse system, where the coupling is given through the first component of the Bresse model with the heat conduction of second sound type. We state the well‐posedness and show the polynomial stability of the system, where the decay rate depends on the smoothness of initial data. Moreover, we prove the non exponential and the exponential decay depending on new conditions on the parameters of the system. The proof is based on a combination of the energy method and the frequency domain approach.

Determining the Coefficients of the Thermoelastic System from Boundary Information

Given a compact Riemannian manifold (M,g) with smooth boundary ∂M, we give an explicit expression for the full symbol of the thermoelastic Dirichlet-to-Neumann map Λg with variable coefficients λ,μ,α,β∈C∞(M¯). We prove that Λg uniquely determines partial derivatives of all orders of these coefficients on the boundary ∂M. Moreover, for a nonempty smooth open subset Γ⊂∂M, suppose that the manifold and these coefficients are real analytic up to Γ. We show that Λg uniquely determines these coefficients on the whole manifold M¯.

On a thermoelastic Timoshenko system without the second spectrum: Existence and stability

In this work we study the well-posedness and the asymptotic stability of a linear thermoelastic Timoshenko system free of second spectrum, where the heat conduction is given by Cattaneo’s law. We give a detailed proof of the well-posedness, using the semigroup theory. Then, we prove that the system is exponentially stable irrespective of the coefficients. We also illustrate our theoretical results by performing some numerical tests.

Exponential stability for a porous thermoelastic system of type II

In this paper, we study the stability of a porous thermoelastic system of type II. When a global viscous dissipation is imposed on the elastic wave equation or two local viscous dissipations are imposed on the elastic wave equation and the wave equation of the volume fraction, respectively, we show that the system is exponentially stable by the frequency domain method.

Stability of thermoelastic Timoshenko beam with suspenders and time-varying feedback

This paper considers a one-dimensional thermoelastic Timoshenko beam system with suspenders, general weak internal damping with time varying coefficient, and time-varying delay terms. Under suitable conditions on the nonlinear terms, we prove a general stability result for the beam model, where exponential and polynomial decay are special cases. We also gave some examples to illustrate our theoretical finding.

Non-exponential stability to a Timoshenko system with heat conduction and Kelvin–Voigt damping

We consider a Timoshenko system with partially Kelvin–Voigt damping, and with Cattaneo/Fourier type heat conduction. When the Timoshenko system with only heat conduction or only partially Kelvin–Voigt damping, some authors showed the non-exponential stability. In this paper, we add the Kelvin–Voigt damping in the thermoelastic Timoshenko system, and prove that the system is also not exponentially stable in cases of Fourier and Cattaneo type heat conduction, whether the wave speeds are equal or not. Additionally, we prove the semigroup decays with the optimal rate t−12.

Global existence and exponential decay for thermoelastic system with nonlinear distributed delay

In this work, we consider the thermoelastic system with a nonlinear distributed delay. Under appropriate hypothesis on the weight function of the delay we prove well-posedness by using the Feado- Galerkin method. Also we establish an exponential decay result by introducing a suitable Lyapunov functional.

Exponential stability of Thermoelastic system with boundary time-varying delay

<abstract><p>In this paper, we consider a Thermoelastic system with boundary time-varying delay. Under some appropriate assumptions, the global well-posedness and exponential stability are obtained by using the variable norm technique of Kato and the energy method respectively.</p></abstract>

Long-time behavior for a nonlinear Timoshenko system: Thermal damping versus weak damping of variable-exponents type

<abstract><p>In this work, we consider a nonlinear thermoelastic Timoshenko system with a time-dependent coefficient where the heat conduction is given by Coleman-Gurtin <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. Consequently, the Fourier and Gurtin-Pipkin laws are special cases. We prove that the system is exponentially and polynomially stable. The equality of the wave speeds is not imposed unless the system is not fully damped by the thermoelasticity effect. In other words, the thermoelasticity is only coupled to the first equation in the system. By constructing a suitable Lyapunov functional, we establish exponential and polynomial decay rates for the system. We noticed that the decay sometimes depends on the behavior of the thermal kernel, the variable exponent, and the time-dependent coefficient. Our results extend and improve some earlier results in the literature especially the recent results by Fareh <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>, Mustafa <sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup> and Al-Mahdi and Al-Gharabli <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>.</p></abstract>

Internal exact-approximate controllability for the thermoelastic Bresse system

Internal exact-approximate controllability for the thermoelastic Bresse system

A numerical study of swelling porous thermoelastic media with second sound

In this work, we numerically consider a swelling porous thermoelastic system with a heat flux given by the Maxwell–Cattaneo law. We study the numerical energy and the exponential decay of the thermoelastic problem. First, we give a variational formulation written in terms of the transformed derivatives corresponding to a coupled linear system composed of four first-order variational equations. A fully discrete algorithm is introduced and a discrete stability property is proven. A priori error estimates are also provided. Finally, some numerical results are given to demonstrate the behavior of the solution.