Exponential decay to a thermoelastic system with indefinite memory dissipation in a domain with moving boundary

We study the thermoelastic system in a domain with moving boundary, which was obtained when, instead of the Fourier’s law for the heat flux relation, we followed the linearized model proposed by Coleman and Gurtin [3] and Gurtin and Pipkin [6] about the memory theory of heat conduction. We show the existence, uniqueness and exponential decay rate of global regular solutions.     

Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics

In the present paper, we study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the equation is an abstract hyperbolic equation perturbed by summands with Volterra integral operators. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media (the Darcy law). The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis. Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.     

A thermoelastic system of memory type in noncylindrical domains

In this paper, we consider a hyperbolic thermoelastic system of memory type in domains with moving boundary. The problem models vibrations of an elastic bar under thermal effects according to the heat conduction law of Gurtin and Pipkin. Global existence is proved by using the penalty method of Lions.     

Attractor for a conserved phase‐field system with hyperbolic heat conduction

We consider a conserved phase‐field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The latter dependence is a law of Gurtin–Pipkin type, so that the equation ruling the temperature evolution is hyperbolic. Thus, the system consists of a hyperbolic integrodifferential equation coupled with a fourth‐order evolution equation for the phase‐field. This model, endowed with suitable boundary conditions, has already been analysed within the theory of dissipative dynamical systems, and the existence of an absorbing set has been obtained. Here we prove the existence of the universal attractor. Copyright © 2004 John Wiley & Sons, Ltd.     

Exact controllability of thermoelastic plate equation with memory

In this paper, we consider exact control problem for a coupled system of plate with Gurtin‐Pipkin equation. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the dual system. Firstly, we obtain the observability inequality of the dual system by means of multiplier method. Then, we prove that the system is exactly controllable based on the Hilbert Unique Method.     

Wave propagation aspects of the generalized theory of heat conduction

Summary It is well known that the classical theory of heat conduction, which is based upon Fourier's law, leads to infinite propagation speeds for thermal disturbances. In a recent investigation [1], Gurtin and Pipkin devised a theory appropriate to rigid heat conductors with memory, and put forth evidence that their theory gives rise in general to finite wave speeds. The present paper is concerned with the linearized version of the theory presented in [1], in the form it assumes for isotropic conductors. We arrive at conditions upon the material response functions that ensure the finiteness of the wave speeds. In addition, we establish uniqueness of solutions for a class of history-value problems suggested by the linearized theory.

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Zusammenfassung Bekanntlich führt Fourier's klassische Theorie der Wärmeleitfähigkeit zu einer unendlich großen Ausbreitungsgeschwindigkeit lokaler Temperaturstörungen. Gurtin und Pipkin haben eine Theorie für starre Wärmeleiter mit Gedächtnis eingeführt und haben auch einen Beweis dafür gegeben, daß ihre Theorie auf eine endliche Ausbreitungsgeschwindigkeit führt. Die vorliegende Arbeit bezieht sich auf die linearisierte Form der Theorie von Gurtin und Pipkin für isotrope Leiter. Es werden Bedingungen für endliche Ausbreitungsgeschwindigkeit angegeben. Ferner wird die Eindeutigkeit der Lösungen für eine Klasse von history-value-Problemen angegeben, die durch die lineare Theorie nahegelegt werden.

     

Propagation of nonlinear thermoelastic waves in porous media within the theory of heat conduction with memory: physical derivation and exact solutions

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid‐saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time‐fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time‐fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Blow-up of solutions to the Cauchy problem in nonlinear one-dimensional thermoelasticity

This paper is concerned with the blow-up phenomena of solutions to the Cauchy problem in non-autonomous nonlinear one-dimensional thermoelastic models obeying both Fourier's law of heat flux and the theory due to Gurtin and Pipkin. Moreover some previously related results have been extended.     

Regularity and convergence results for a phase–field model with memory

A phase–field model based on the Coleman–Gurtin heat flux law is considered. The resulting system of non-linear parabolic equations, associated with a set of initial and Neumann boundary conditions, is studied. Existence, uniqueness, and regularity results are proved. An asymptotic analysis is also carried out, in the case where the coefficient of the interfacial energy term tends to 0. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Splitting schemes for hyperbolic heat conduction equation

Rapid processes of heat transfer are not described by the standard heat conduction equation. To take into account a finite velocity of heat transfer, we use the hyperbolic model of heat conduction, which is connected with the relaxation of heat fluxes. In this case, the mathematical model is based on a hyperbolic equation of second order or a system of equations for the temperature and heat fluxes. In this paper we construct for the hyperbolic heat conduction equation the additive schemes of splitting with respect to directions. Unconditional stability of locally one-dimensional splitting schemes is established. New splitting schemes are proposed and studied for a system of equations written in terms of the temperature and heat fluxes.     

Stability results of a distributed problem involving Bresse system with history and/or Cattaneo law under fully Dirichlet or mixed boundary conditions

In this paper, we study the stability of a 1‐dimensional Bresse system with infinite memory‐type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. When the thermal effect vanishes, the system becomes elastic with memory term acting on one equation. We consider the interesting case of fully Dirichlet boundary conditions. Indeed, under equal speed of propagation condition, we establish the exponential stability of the system. However, in the natural physical case when the speeds of propagation are different, using a spectrum method, we show that the Bresse system is not uniformly exponentially stable. In this case, we establish a polynomial energy decay rate. Our study is valid for all other mixed boundary conditions.     

Spatial bounds and growth estimates for the heat equation with three relaxation times

In this note we investigate spatial bounds for the heat conduction equation with three relaxation times. We use energy methods similar to those considered in the study of the backward parabolic heat equation. Finally, we show that for any positive choice of the coefficients we may take initial data for which the solution is unstable when we suppose that the body is bounded. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A positivity‐preserving nonstandard finite difference scheme for the damped wave equation

A positivity‐preserving nonstandard finite difference scheme is constructed to solve an initial‐boundary value problem involving heat transfer described by the Maxwell‐Cattaneo thermal conduction law, i.e., a modified form of the classical Fourier flux relation. The resulting heat transport equation is the damped wave equation, a PDE of hyperbolic type. In addition, exact analytical solutions are given, special cases are mentioned, and it is noted that the positivity condition is equivalent to the usual linear stability criteria. Finally, solution profiles are plotted and possible extensions to a delayed diffusion equation and nonlinear reaction‐diffusion systems are discussed. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

Heat equation with memory in anisotropic and non-homogeneous media

A modified Fourier’s law in an anisotropic and non-homogeneous media results in a heat equation with memory, for which the memory kernel is matrix-valued and spatially dependent. Different conditions on the memory kernel lead to the equation being either a parabolic type or a hyperbolic type. Well-posedness of such a heat equation is established under some general and reasonable conditions. It is shown that the propagation speed for heat pulses could be either infinite or finite, depending on the different types of the memory kernels. Our analysis indicates that, in the framework of linear theory, heat equation with hyperbolic kernel is a more realistic model for the heat conduction, which might be of some interest in physics.     

Thermoelasticity that uses fractional heat conduction equation

A survey of nonlocal generalizations of the Fourier law and heat conduction equation is presented. More attention is focused on the heat conduction with time and space fractional derivatives and on the theory of thermal stresses based on this equation.     

On the decay rates of Timoshenko system with second sound

In this work, we study the well‐posedness and the asymptotic stability of a one‐dimensional linear thermoelastic Timoshenko system, where the heat conduction is given by Cattaneo's law and the coupling is via the displacement equation. We prove that the system is exponentially stable provided that the stability number χ_τ=0. Otherwise, we show that the system lacks exponential stability. Furthermore, in the latter case, we show that the solution decays polynomially. Copyright © 2015 John Wiley & Sons, Ltd.     

Integro-differential equation modelling heat transfer in conducting,radiating and semitransparent materials

In this work we analyse a model for radiative heat transfer in materials that are conductive, grey and semitransparent. Such materials are for example glass, silicon, water and several gases. The most important feature of the model is the non-local interaction due to exchange of radiation. This, together with non-linearity arising from the well-known Stefan–Boltzmann law, makes the resulting heat equation non-monotone. By analysing the terms related to heat radiation we prove that the operator defining the problem is pseudomonotone. Hence, we can prove the existence of weak solution in the cases where coercivity can be obtained. In the general case, we prove the solvability of the system using the technique of sub and supersolutions. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On the stability of the equilibrium states for Hamiltonian dynamical systems arising in non–equilibrium thermodynamics

In this paper we consider a thermodynamic system with an internal state variable, and study the stability of its equilibrium states by exploiting the reduced entropy inequality. Remarkably, we derive a Hamiltonian dynamical system ruling the evolution of the system in a suitable thermodynamic phase space. The use of the Hamiltonian formalism allows us to prove the equivalence of the asymptotic stability at constant temperature, at constant entropy and at constant energy, thus extending some classical results by Coleman and Gurtin (J. Chem. Phys., 47, 597–613, 1967).     

Longtime dynamics for a novel piezoelectric beam model with creep and thermo-viscoelastic effects

In this work, a novel fully-dynamic piezoelectric beam model is considered. Electromagnetic and thermal effects are taken into consideration by Maxwell’s equations and the Coleman–Gurtin law (instead of the Fourier’s law), respectively. Our model accounts also for thermal and electromagnetic (creep) past histories, which are in line with the time response of PVDF at the applied stress in the longitudinal direction. Under suitable assumptions, the existence and uniqueness of solutions are proved by the semigroup theory. The main purpose of this paper is to establish the longtime dynamics of the model. Therefore, the quasi-stability property of the model and the existence of smooth global attractors with finite fractal dimension are obtained. The existence of exponential attractors for the associated dynamical system is also proved.     

Nonlinear damped Timoshenko systems with second sound—Global existence and exponential stability

In this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one‐dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle—a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneo's law, respectively. The global existence of small, smooth solutions and the exponential stability in linear and nonlinear cases are established. Copyright © 2008 John Wiley & Sons, Ltd.