Antonio Magaña

It is well known that in classical thermoelasticity, there is no direct coupling between displacement and microtemperatures. However, this situation changes when considering the Type Ⅱ and Type Ⅲ theories of Green and Naghdi. Furthermore, it is established that in the Type Ⅲ theory, dissipation at the microtemperature level can lead to the exponential decay of the 3-dimensional system. Recently, second-gradient heat conduction theories have been introduced, making it natural to wonder whether the behavior observed in first-gradient theories persists. In this paper, we consider a 3-dimensional viscoelastic solid with microtemperatures (both of the second-gradient type). We introduce dissipative mechanisms into the mechanical part (Kelvin-Voigt first and Zener later), while the microtemperature structure remains conservative. We demonstrate that, under fairly natural hypotheses, we can ensure the exponential decay of the entire system. That is, we prove that the coupling in this case is also strong.

We discuss the strategic situation in the Barcelona Town Council after the elections held on May 28, 2023. This institution is described as a simple game, and the Shapley–Shubik index is applied to measure the initial power distribution among the parties. We introduce some ideological incompatibilities in the game and, finally, we discuss the existence of stable coalition structures in the different stated scenarios when the power payoffs to the players are computed using the proportional partitional Shapley value. The key point is the formation of a stable coalition structure. Broadly speaking, a coalition structure is stable if the payoff obtained by each player cannot be increased by a unilateral change in their choice of coalition. In a previous paper, we showed that any simple game admits stable coalition structures when the payoffs are computed using the proportional partitional Shapley value. We find the stable coalition structures for each of the situations analyzed. The results show a fairly complicated problem that the involved politicians and parties should try to solve for the benefit of Barcelona citizens.

In this work we analyse the time decay of solutions for the second gradient thermoelasticity when two dissipation mechanisms are introduced in the system. First we consider that the fourth spatial derivative of the velocity is present in the dissipation and prove that the decay of the solutions cannot be of exponential type. In fact, we show that the solutions decay polynomially, and in two different ways depending on how the variables are coupled. Second we consider only the Laplacian of the velocity in the dissipation and we prove that the solutions decay exponentially. The lack of analyticity of the semigroup is also shown. The mathematical tools we use are the Lumer–Phillips corollary to the Hille–Yosida theorem, the characterization of the exponentially stable semigroups provided by Huang–Prüss and the one corresponding to polynomially stable semigroups obtained by Borichev–Tomilov.

We study from a numerical point of view a multidimensional problem involving a viscoelastic body with two porous structures. The mechanical problem leads to a linear system of three coupled hyperbolic partial differential equations. Its corresponding variational formulation gives rise to three coupled parabolic linear equations. An existence and uniqueness result, and an energy decay property, are recalled. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some numerical simulations are performed in one and two dimensions to show the accuracy of the approximation and the behaviour of the solution. Viscoelasticity with double porosity, Finite elements, A priori estimates, Numerical simulations. 65M60, 37N15, 74F05, 65M12 COPYRIGHT: © Global Science Press noabaza@hotmail.com (Noelia Bazarra) jose.fernandez@uvigo.es (José R. Fernández) Mari.Carme.Leseduarte@upc.edu (Mari Carme Leseduarte) antonio.magana@upc.edu (Antonio Magaña) Ramon.Quintanilla@upc.edu (Ramón Quintanilla)

In this work we analyze, from the numerical point of view, a dynamic problem involving a thermoelastic rod. Two porosities are considered: the first one is the macro-porosity, connected with the pores of the material, and the other one is the micro-porosity, linked with the fissures of the skeleton. The mechanical problem is written as a set of hyperbolic and parabolic partial differential equations. An existence and uniqueness result and an energy decay property are stated. Then, a fully discrete approximation is introduced using the finite element method and the backward Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some numerical simulations are presented to show the behaviour of the approximation.