Journal ArticleDOI
Decay of solutions of the system of thermoelasticity of type iii
Xu Zhang,Xu Zhang,Enrique Zuazua +2 more
TLDR
In this paper, the long time behavior of solutions of the system of thermoelasticity of type III in a bounded domain of ℝn (n = 1,2,3) and in the whole space of n is analyzed.Abstract:
This paper is devoted to analyzing the long time behavior of solutions of the system of thermoelasticity of type III in a bounded domain of ℝn (n = 1,2,3) and in the whole space ℝn. For the first case, we introduce a decoupled system that allows to reduce the problem of the asymptotic behavior for the original system to a suitable observability inequality for the Lame system. In this way most of the existing results for the classical system of thermoelasticity are shown to hold for this system too. In particular, we show that: (1) For most domains the energy of the system does not decay uniformly; (2) Under suitable conditions on the domain that may be described in terms of Geometric Optics, the energy of the system decays exponentially; and (3) For most domains in two space dimensions, the energy of smooth solutions decays polynomially. For the problem in the whole space ℝn, first, based on Fourier analysis and Lyapunov's second method, we show that the energy of longitudinal and thermal waves of the system decays as that of the classical heat equation (while that of the transversal wave component is conservative). Then, by means of a careful spectral analysis, we give a sharp description on the decay rate of the high frequency longitudinal and thermal waves of the system.read more
Citations
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Journal ArticleDOI
Exponential stability in thermoelasticity with microtemperatures
Pablo S. Casas,Ramón Quintanilla +1 more
TL;DR: In this article, a linear theory for elastic materials with inner structure, whose particles in addition to the classical displacement, possess microtemperatures, is presented. And the exponential stability of the solutions when considering the theory with microtemperature is proved.
Journal ArticleDOI
Exponential decay in one-dimensional porous-thermo-elasticity
Pablo S. Casas,Ramón Quintanilla +1 more
TL;DR: In this paper, the authors considered two kinds of dissipation process: the viscosity type in the porous structure and the thermal dissipation, and they proved that when both kinds of terms are taken into account in the evolution equations the solutions are exponentially stable.
Journal ArticleDOI
Exact Controllability for Multidimensional Semilinear Hyperbolic Equations
Xiaoyu Fu,Jiongmin Yong,Xu Zhang +2 more
TL;DR: A global exact controllability result is obtained for a class of multidimensional semilinear hyperbolic equations with a superlinear nonlinearity and variable coefficients, in which the crucial observability constant is estimated explicitly by a function of the norm of the potential.
Journal ArticleDOI
Energy decay in a Timoshenko-type system of thermoelasticity of type III
TL;DR: In this article, the authors consider a one-dimensional linear thermoelastic system of Timoshenko type, where the heat conduction is given by Green and Naghdi theories, and prove the exponential stability by using the energy method.
Journal ArticleDOI
Energy decay for Porous-thermo-elasticity systems of memory type
TL;DR: In this article, a memory term in one dimension was proposed and an exponential and polynomial decay result was established for the memory term with respect to the exponential decay of the memory.
References
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Semigroups of Linear Operators and Applications to Partial Differential Equations
TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
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TL;DR: In this paper, the authors consider the problem of finding solutions to elliptic boundary value problems in Spaces of Analytic Functions and of Class Mk Generalizations in the case of distributions and Ultra-Distributions.
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Interpolation Spaces: An Introduction
Jöran Bergh,Jörgen Löfström +1 more
TL;DR: In this paper, the authors define the Riesz-Thorin Theorem as a necessary and sufficient condition for interpolation spaces, and apply it to approximate spaces in the context of vector spaces.
Journal ArticleDOI
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary
TL;DR: For the observation or control of solutions of second-order hyperbolic equation in this paper, Ralston's construction of localized states [Comm. Pure Appl. Math, 22 (1969), pp.
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On undamped heat waves in an elastic solid
Albert Edward Green,P. M. Naghdi +1 more
TL;DR: In this article, the authors focused on the thermal properties of the constitutive response functions in the context of both nonlinear and linear theories, and provided an easy comparison of the one-dimensional version of the equation for the determination of temperature in the linearized theory.