citable using Digital Object Identifier – DOI) Early View publication on wileyonlinelibrary.com (issue and page numbers not yet assigned; ZAMM · Z. Angew. Math. Mech., 1 – 16 (2013) / DOI 10.1002/zamm.201300050 A theory of thermoelasticity with diffusion under Green-Naghdi models 1 Moncef Aouadi 1, ∗ , Barbara Lazzari 2, ∗∗ , and Roberta Nibbi 2, ∗∗∗ 2 1 Department of Mathematics and Computer Science, Institut Sup´ erieur des Sciences Appliqu´ ees et de Technologie de 3 Mateur, University of Carthage, Tunisia 4 2 Department of Mathematics, University of Bologna, 5 Piazza di Porta S. Donato, 40126 Bologna, Italy 5 Received 26 February 2013, revised 25 May 2013, accepted 8 June 2013 6 Published online ♣ 2013 7 Key words Thermoelastic diffusion, Green-Naghdi theory, well-posedness, asymptotic behavior, localization in time. 8 In this paper, we use the Green-Naghdi theory of thermomechanics of continua to derive a nonlinear theory of thermoe- lasticity with diffusion of types II and III. This theory permits propagation of both thermal and diffusion waves at finite speeds. The equations of the linear theory are also obtained. With the help of the semigroup theory of linear operators we establish that the linear anisotropic problem is well posed and we study the asymptotic behavior of the solutions. Finally, we investigate the impossibility of the localization in time of solutions. c  2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction 9 The usual theory of heat conduction based on the Fourier’s law allows the phenomena of ”infinite diffusion velocity” which 10 is not well accepted from a physical point of view. This is referred to as the paradox of heat conduction. It is physically 11 unrealistic since it implies that thermal signals propagate with infinite speed. The articles of Dreyer and Struchtrup [1] and 12 Caviglia et al. [2] provide an extensive survey of works on experiments involving the propagation of heat as a thermal wave. 13 They report instances where the phenomena of second sound has been observed in several kinds of materials. This kind of 14 fact has provoked intense activity in the field of heat propagation. 15 In contrast to the conventional thermoelasticity, nonclassical theories came into existence during the last two decades. 16 These theories, referred to as generalized thermoelasticity, were introduced in the literature in an attempt to eliminate 17 the shortcomings of the classical dynamical thermoelasticity. A survey article of representative theories in the range of 18 generalized thermoelasticity is due to Hetnarski and Ignaczak [3]. 19 Green-Naghi [4, 5] developed a thermomechanical theory of deformable continua that relies on an entropy balance law 20 rather than an entropy inequality, where the heat conduction does not agree with the usual one (see also [6]). However, we 21 want to mention the total compatibility of the entropy balance law with the entropy inequality. They proposed the use of 22 the thermal displacement 23 α(x,t)=  t t0 θ(x,s)ds + α 0 , 24 where θ is the empirical temperature, and considered three theories labelled as type I, II, and III, respectively. These theories 25 were based on an entropy balance law rather than the usual entropy inequality. The type I thermoelasticity coincides with 26 the classical one; in type II, the heat is allowed to propagate by means of thermal waves but without dissipating energy 27 and, for this reason, it is also known as thermoelasticity without energy dissipation. The heat equation of type III, where 28 the heat flux is a combination of type I and II, contains both type I and II as limiting cases. In addition, the thermoelasticity 29 of type III allows the constitutive functions for free energy, stress tensor, entropy and heat flux to depend on the strain 30 tensor, the time derivative of the thermal displacement, the gradient of thermal displacement and the time derivative of 31 the gradient of thermal displacement. This theory allows the dissipation energy, but the heat flux is partially determined 32 from the Helmholtz free energy potential. Both, type II and III, overcome the unnatural property of Fourier’s law of infinite 33 propagation speed and imply a finite wave propagation. 34 All these theories have recently been the subject of great amount of works (as a matter of illustration see [7–14]). 35 ∗ Corresponding author E-mail: moncef aouadi@yahoo.fr ∗∗ E-mail: barbara.lazzari@unibo.it ∗∗∗ E-mail: roberta.nibbi@unibo.it c  2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim