keywords: thermoelasticity with microtemperatures, exponential stability.Running head: Exponential stability in thermoelasticity. AbstractThis article is concerned with a linear theory for elastic materials with inner structure, whose particles in addition to the classical displacement, possess microtemperatures. In the main part of the paper we restrict our attention to the one-dimensional problem. First, we prove the slow decay of solutions for the onedimensional problem of micromorphic elastic solids with the usual thermal effects. Then, we prove the exponential stability of the solutions when we consider the theory with microtemperatures. The anti-plane distributions of microtemperatures are considered later.

This paper concerns the one dimensional problem of the porous-thermo-elasticity. Two kinds of dissipation process are considered: the viscosity type in the porous structure and the thermal dissipation. It is known that when only thermal damping is considered or when only porous damping is considered we have the slow decay of the solutions. Here we prove that when both kinds of dissipation terms are taken into account in the evolution equations the solutions are exponentially stable.

Abstract. The billiard motion inside an ellipsoid Q ⊂ R n+1 is completely integrable. Its phase space is a symplectic manifold of dimension 2n, which is mostly foliated with Liouville tori of dimension n. The motion on each Liouville torus becomes just a parallel translation with some frequency ω that varies with the torus. Besides, any billiard trajectory inside Q is tangent to n caustics Q λ1 , . . . , Q λn , so the caustic parameters λ = (λ 1 , . . . , λ n ) are integrals of the billiard map. The frequency map λ → ω is a key tool to understand the structure of periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the caustic parameters.We present two conjectures, fully supported by numerical experiments. We obtain, from one of the conjectures, some lower bounds on the periods. These bounds only depend on the type of the n caustics. We describe the geometric meaning, domain, and range of ω. The map ω can be continuously extended to singular values of the caustic parameters, although it becomes "exponentially sharp" at some of them.Finally, we study triaxial ellipsoids of R 3 . We compute numerically the bifurcation curves in the parameter space on which the Liouville tori with a fixed frequency disappear. We determine which ellipsoids have more periodic trajectories. We check that the previous lower bounds on the periods are optimal, by displaying periodic trajectories with periods four, five, and six whose caustics have the right types. We also give some new insights for ellipses of R 2 .

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We classify nonsingular symmetric periodic trajectories (SPTs) of billiards inside ellipsoids of R n+1 without any symmetry of revolution. SPTs are defined as periodic trajectories passing through some symmetry set. We prove that there are exactly 2 2n (2 n+1 − 1) classes of such trajectories. We have implemented an algorithm to find minimal SPTs of each of the 12 classes in the 2D case (R 2 ) and each of the 112 classes in the 3D case (R 3 ). They have periods 3, 4 or 6 in the 2D case; and 4, 5, 6, 8 or 10 in the 3D case. We display a selection of 3D minimal SPTs. Some of them have properties that cannot take place in the 2D case.Smooth convex billiards are a paradigm of conservative dynamics, in which a particle collides with a fixed closed smooth convex hypersurface of R n+1 . They provide examples of different dynamics: integrable, mostly regular, chaotic, etc. In this paper we tackle the integrable situation. Concretely, we find and classify symmetric periodic trajectories (SPTs) inside ellipsoids of R n+1 . PTs show different dynamics, which describe how they fold in R n+1 . STs present symmetry with respect to a coordinate subspace of R n+1 . Dynamics and symmetry are precisely the main aspects we consider in our classification of SPTs. We establish 112 classes of SPTs in the 3D case, and we find a representative of each class with the smallest possible period. Those minimal SPTs have periods 4, 5, 6, 8, or 10. We depict a selection of minimal 3D SPTs. Some of them have properties that cannot take place in the 2D case. SPTs are preserved under symmetric deformations of the ellipsoid. In a future paper we plan to study their bifurcations and the transition between stability and instability under such deformations.