International Journal of Mathematics and Mathematical Sciences

Volume 2015, Article ID 479267, 11 pages

http://dx.doi.org/10.1155/2015/479267

## Solving the Linear 1D Thermoelasticity Equations with Pure Delay

^{1}Department of Cybernetics, Kyiv National Taras Shevchenko University, 64 Volodymyrska Street, Kyiv 01601, Ukraine^{2}Department of Mathematics and Statistics, University of Konstanz, Universitätsstraße 10, 78457 Konstanz, Germany

Received 27 October 2014; Accepted 3 January 2015

Academic Editor: Harvinder S. Sidhu

Copyright © 2015 Denys Ya. Khusainov and Michael Pokojovy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a system of partial differential equations with a single constant delay describing the behavior of a one-dimensional thermoelastic solid occupying a bounded interval of . For an initial-boundary value problem associated with this system, we prove a well-posedness result in a certain topology under appropriate regularity conditions on the data. Further, we show the solution of our delayed model to converge to the solution of the classical equations of thermoelasticity as . Finally, we deduce an explicit solution representation for the delay problem.

#### 1. Introduction

Over the past half-century, the equations of thermoelasticity have drawn a lot of attention from the side of both mathematical and physical communities. Starting with the late 50s and early 60s of the last century, the necessity of a rational physical description for elastic deformations of solid bodies accompanied by thermal stresses motivated the more prominent mathematicians, physicists, and engineers to focus on this problem (see, e.g., [1, 2]). As a consequence, many theories emerged, mainly in the cross-section of (nonlinear) field theory and thermodynamics, making it possible for the equations of thermoelasticity to be interpreted as an anelastic modification of the equations of elasticity (cf. [3] and the references therein). Both linear and nonlinear models and solution theories were proposed.

An initial-boundary value problem for the general linear equations of classical thermoelasticity in a bounded smooth domain was studied by Dafermos in [4]. Here, and denote the (unknown) displacement vector field and the absolute temperature, respectively. Further, is the material density, is a reference temperature rendering the body free of thermal stresses, is the specific heat capacity, stands for Hooke’s tensor, is the stress-temperature tensor, is the heat conductivity tensor, represents the specific external body force, and is the external heat supply. Under usual initial conditions, appropriate normalization conditions to rule out the rigid motion as a trivial solution, and general boundary conditions where are relatively open, denotes the “elasticity” modulus, and is heat transfer coefficient, Dafermos proved the global existence and uniqueness of finite energy solutions and studied their regularity as well as asymptotics as . In 1D, even an exponential stability result for (1)-(2) under all “reasonable” boundary conditions was shown by Hansen in [5].

In his work [6], Slemrod studied the nonlinear equations of 1D thermoelasticity in the Lagrangian coordinates for the unknown functions denoting the displacement of the rod and being a temperature difference to a reference temperature rendering the body free of thermal stresses. The functions and denote the Helmholtz free energy and the heat flux, respectively, and are assumed to be given. Finally, is the material density in the references configuration. Under appropriate boundary conditions (when the boundary is free of tractions and is held at a constant temperature or when the body is rigidly clamped and thermally insulated) as well as usual initial conditions for both unknown functions, a local existence theorem for (4) was proved by additionally imposing a regularity and compatibility condition. For sufficiently small initial data, the local classical solution could be globally continued. At the same time, when studying (4) in the whole space, large data are known to lead to a blow-up in final time (cf. [7]).

Racke and Shibata studied in [8] (4) under homogeneous Dirichlet boundary conditions for both and . Under appropriate smoothness assumptions, they proved the global existence and exponential stability for the classical solutions to the problem. In contrast to Slemrod [6], their method was using spectral analysis rather than ad hoc energy estimates obtained by differentiating the equations with respect to and . A detailed overview of further recent developments in the field of classical thermoelasticity and corresponding references can be found in the monograph [9] by Jiang and Racke.

The classical equations of thermoelasticity outlined above, being a hyperbolic-parabolic system, provide a rather good macroscopic description in many real-world applications. At the same time, they sometimes fail when being used to model thermoelastic stresses in some other situations, in particular, in extremely small bodies exposed to heat pulses of large amplitude (see, e.g., [10]) and so forth. To address these issues, a new theory, commonly referred to as the theory of hyperbolic thermoelasticity or second sound thermoelasticity, has emerged. In contrast to the classical thermoelasticity, parabolic equation (2) is replaced with a hyperbolic first-order system with and denoting the heat flux and the relaxation tensor, respectively. Both linear and nonlinear versions of the equations of hyperbolic thermoelasticity (1), (5)-(6) have been studied in the literature. See, for example, [11] by Messaoudi and Said-Houari for a proof of global well-posedness of the 1D system in the whole space or Irmscher’s work [12] for the global well-posedness of nonlinear problem for rotationally symmetric data in a bounded rotationally symmetric domain of . In a bounded 1D domain, a quantitative stability comparison between the classical and the hyperbolic system was presented by Irmscher and Racke in [13]. For a detailed overview on hyperbolic thermoelasticity, we refer the reader to [14] by Chandrasekharaiah and [15] by Racke.

A unified approach establishing a connection between the classical and hyperbolic thermoelasticity was developed by Tzou in [16, 17]. Namely, he proposed to view (6) with as a first-order Taylor approximation of the equation being equivalent to the delay equation More generally, higher-order Taylor expansion to the dual phase lag constitutive equation can be considered. Together with (1)-(2), (5), this leads to the so-called dual phase lag thermoelasticity studied by Racke (cf. references in [15, page 415]).

If no Taylor expansion with respect to is carried out in (8), it can be shown that the corresponding system is ill-posed when being considered in the same topology as the original system of classical thermoelasticity (cf. [18]); that is, the system is lacking a continuous dependence of solution on the data. Moreover, the delay law (8) can, in general, contradict the second law of thermodynamics as shown in [19].

Nonetheless, it remains desirable to understand the dynamics of equations of thermoelasticity originated from delayed material laws. One of the first attempts to obtain a well-posedness result for a partial differential equation with pure delay is due to Rodrigues et al. In their paper [20], Rodrigues et al. studied a heat equation with pure delay in an appropriate Frechét space and showed the delayed Laplacian to generate a -semigroup on this space. Further, they investigated the spectrum of the infinitesimal generator. Though their approach can essentially be carried over to the equations of thermoelasticity with pure delay derived in Section 2, we propose a new approach in this paper preserving the Hilbert space structure of the space and thus the connection to the classical equations of thermoelasticity. To the authors’ best knowledge, no results on thermoelasticity with delay in the highest order terms have been previously published in the literature. At the same time, we refer the reader to the works by Khusainov et al. [21–24], in which the authors studied the well-posedness and controllability for the heat and/or the wave equation on a finite time horizon. In their recent paper [25], Khusainov et al. exploited the -maximum regularity theory to prove a global well-posedness and asymptotic stability results for a regularized heat equation with delay.

The present paper has the following outline. In Section 2, we give a physical model for linear thermoelasticity based on delayed material laws. For the sake of simplicity, we present a 1D model though our approach can easily be carried over to the general multidimensional case. Next, in Section 3, we prove the well-posedness of this model in an appropriate Hilbert space framework and discuss the small parameter asymptotics, that is, the behavior of solutions as . Further, in Section 4, we deduce an explicit solution representation formula. Finally, in the Appendix, we summarize some seminal results on the delayed exponential function and Cauchy problems with pure delay.

#### 2. Model Description

We consider a solid body occupying an axis-aligned rectangular domain of . Assuming that the body motion is purely longitudinal with respect to the first space variable (cf. [6, page 100]), deformation gradient, stress and strain tensors, and so forth are diagonal matrices and a complete rational description of the original 3D body motion can be reduced to studying the 1D projection , , of the body onto the -axis as displayed in Figure 1. Hence, in the following, we restrict ourselves to considering the relevant physical values only in -direction.