Asymptotic behaviour for a two-dimensional thermoelastic model M. Fabrizio, B. Lazzari, J. M. Rivera Abstract In this paper we study a thermoelastic material with an internal struc- ture which binds the materials fibres to a quadratic behavior. Moreover a hereditary constitutive law for heat flux is supposed. We prove results of asymptotic stability and exponential decay for the evolution problem in two-dimensional space domain. MOS Classification: 35L45 45K05 74D99 1 Introduction The thermoelastic system, with conservative boundary condition in general is not exponentially stable for materials configurated in two or three dimensional space. This is because the dissipative component due to heat flux is not enough to produce the uniform rate of decay (see [7]), which in particular means that zero is not in general the equilibrium point. Of course this is not the case for materials configurated over one dimensional sets see for example [1, 5, 9]. In this work we consider a particular thermoelastic system in two dimensional space with an internal structure that considers the elastic fibres following a quadratic law. The equations that model the oscillations of this type of problem is written as follows ¨ w − a(Δw − χw)+ α(λΔτ − τ )=0 in Ω×]0, ∞[ (1.1) c ˙ τ − α(λΔ˙ w − ˙ w)+ στ − σ ′ ∗ τ − k ∗ (δΔτ − τ )=0 in Ω×]0, ∞[ (1.2) where by w we denote the displacement, τ is the difference of temperature. In this model we have considered the flux law of memory type. The equations (1.1)–(1.2) constitute a dissipative system, where the dissipation is given by heat flux and by the hereditary effect of the temperature. In general, for thermoelastic problem, the coupling term is not sufficient to transmit the thermal dissipation to elastic system. Our model presents a second order coupling term which is sufficient to obtain a global dissipation for the system. Moreover we have considered a hereditary constitutive law for heat flux. Unlike Fourier Law, this model is compatible with a finite propagation 1