1R48. Evolution Equations in Thermoelasticity. Monographs and Surveys in Pure and Applied Mathematics, Vol 112. - Song Jiang (Inst of Appl Phys and Comput Math, Beijing, Peoples Rep of China) and E Racke (Univ of Konstanz, Konstanz, Germany). Chapman and Hall/CRC, Boca Raton FL. 2000. 308 pp. ISBN 1-58488-215-8. \$84.95.

Reviewed by MV Shitikova (Dept of Struct Mech, Voronezh State Univ of Architec and Civil Eng, ul Kirova 3-75, Voronezh, 394018, Russia).

The authors’ aim is to present a state of the art in the treatment of initial value problems and of initial boundary value problems both in linear and nonlinear thermoelasticity. From the very beginning, the authors restrict themselves by considering the conventional thermoelasticity theory formulated on the principles of the classical theory of heat conduction, and consequently, the heat transport equation of the theory is of parabolic type. The second grave limitation is connected with the boundary conditions used throughout the monograph, namely: the ideal heat exchange between a thermoelastic body and its surrounding medium and heat insulation on the body’s boundary are considered. The more general condition—condition of heat exchange between thermoelastic bodies or between a body and surrounding medium, from which the conditions of constant temperature or heat insulation follow as particular limiting cases—is not investigated at all. But the condition of heat exchange is the most interesting and important for engineering applications, especially in contact problems. That is why this reviewer cannot agree with the authors that “the intended audience includes not only graduate students of both mathematics and physics, but also the foremost expert looking for a survey.” This book may be useful only for students wanting to be familiar with the basics of mathematical aspects of the convectional thermoelasticity theory.

The book includes nine chapters followed by two appendices, lists of main and supplemental references, notation, and an index. The first chapter gives a short summary of the derivation of the equations describing the nonlinear behavior of a thermoelastic body within the framework of the conventional thermoelasticity theory. Using Taylor expansions, the corresponding linearized equations are also written. The well-posedness of the linear initial boundary value problem in the case of the ideal thermal contact between a rigidly clamped body and surrounding medium is discussed in Chapter 2.

The asymptotic behavior as time tends to infinity of such a thermoelastic system with zero exterior forces and heat supply is investigated in linearized one-dimensional formulations in Chapter 3; two- or three- dimensional formulations are investigated in Chapter 4. A local existence theorem for the initial boundary value problem of hyperbolic-parabolic type and for the Cauchy problem is proved in Chapter 5. One-dimensional and three-dimensional thermoelastic nonlinear equations are considered in Chapters 6 and 7, respectively.

Chapter 8 analyzes the evolution of temperature and displacement in an elastic body that may come into contact with a rigid foundation. The system consists of the linearized equations together with the ideal thermal contact between the body and rigid foundation and Signorini’s nonlinear conditions for mechanical contact. In the final chapter, the following problems are briefly described: the linear boundary value problem in the presence of external forces and heat supply, resulting in an additional damping; the far field asymptotic behavior of the solution, as well as a numerical scheme for the numerical solution of the initial boundary value problem.

Thus, the majority of the book, Chapters 2–8, is devoted to the proof of the obvious results about the exponential damping of energy and displacements as time goes to infinity by the use of different mathematical methods. In this reviewer’s opinion, some results dealing with the asymptotic behavior of the desired values at large times can be obtained by the Laplace transformation methods with the corresponding limiting theorems.

Much to this reviewer’s surprise, the authors, when discussing the behavior of the surfaces of strong discontinuity, did not even mention the works by VI Danilovskaya, RB Hetnarski, J Ignaszak, W Nowacki, and many others who have shown that singularities propagate in the stressed-strained thermoelastic medium of the hyperbolic-parabolic type with damping which has exponential character and is defined by the coupling of the strain and temperature fields. All the researchers mentioned above in one way or another investigated the asymptotic behavior of the solutions obtained as $t→∞$ or $x→∞.$ As for the thermoelastic contact problems, then the authors completely ignore the results by Barber and his coauthors. That is why this reviewer does not share the opinion of the authors that their book “presents a state-of-the-art treatment of initial boundary value problems in thermoelasticity and includes the most extensive bibliographies on the subject published to date.” Quite to the contrary, the lists of main and additional references are very limited and do not cover a huge amount of monographs and original papers dealing with solving the boundary value problems even in the framework of the conventional thermoelasticity theory, not to mention the extended thermoelasticity theories predicting a finite speed of the propagation of thermal signals.

Applications of the mathematical treatment described in the book are of limited usefulness, and this book cannot attract the attention of engineers and researchers involved with a practical implementation of thermoelasticity. This reviewer thinks that Evolution Equations in Thermoelasticity. Monographs and Surveys in Pure and Applied Mathematics, Vol 1 can be useful only for students who want to have some basic mathematical knowledge in classical thermoelasticity, but it cannot be recommended for purchase by libraries for mechanical or civil engineering departments, or by individuals with an interest in the practical utility of thermoelasticity.