Research Article Exponential Decay of Swelling Porous Elastic Soils with Microtemperatures Effects Ali Rezaiguia , 1,2 Salah Zitouni , 2 and Hasan Nihal Zaidi 1 1 Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi Arabia 2 Department of Computer Science and Mathematics, Mouhamed Cherif Messadia University, Souk Ahras, Algeria Correspondence should be addressed to Ali Rezaiguia; ali_rezaig@yahoo.fr Received 11 March 2023; Revised 17 April 2023; Accepted 4 May 2023; Published 13 June 2023 Academic Editor: Genni Fragnelli Copyright © 2023 Ali Rezaiguia et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this article, we considered the one-dimensional swelling problem in porous elastic soils with microtemperatures efects in the case of fuid saturation. First, we showed that the system is well-posed in the sens of semigroup. Ten, we constructed a suitable Lyapunov functional based on the energy method and we proved that the dissipation given only by the microtemperatures is strong enough to provoke an exponential stability for the solution irrespective of the wave speeds of the system. 1. Introduction In [1], Eringen developed a continuum theory for a mixture consisting of three components: an elastic solid, viscous fuid, and gas. Also, the author obtained the feld equations for a heat-conducting mixture. In the theory of mixtures, the great abstraction was extended by assuming that the con- stituents of a mixture could be modeled as superimposed continua, for that each point in the mixture was simulta- neously occupied by a material point of each constituent. A brief description concerning the details of the historical development/review related to the general theory of the mixtures is given by Bedford and Drumheller in [2]. Swelling porous media have been studied in many disparate felds including soil science, hydrology, forestry, geotechnical, chemical, and mechanical engineering, and this is due to its prevalence in nature and modern tech- nologies. In this article, we focused on the asymptotic behavior of swelling soils that belong to the porous media theory in the case of fuid saturation. Swelling soils contain clay minerals that change volume with water content changes that result in major geological hazards and ex- tensive damage worldwide. Te swelling soils are caused by the chemical attraction of water, where water molecules are incorporated in the clay structure in between the clay plates separating and destabilizing the mineral structure. Te swelling clay particles have the property of forming a unit (particle) from lattice hydrated aluminum and magnesium silicate minerals. Tus, the clay’s particle is a mixture of clay platelets and adsorbed water (vicinal water). Such a particle can be thought to defne a mesoscale which is large compared to platelet, but small compared to the soil itself. A proper description of the mesoscale system behavior is critical when modeling consolidation of a swelling clay soil. As pointed out by Eringen [1], this system is the prototype for difusion type models in swelling soils ([3–5]). As established by Ies ¸an [6] and simplifed by Quintanilla [7] (see also [8, 9]), the basic feld equations for the linear theory of swelling porous elastic soils are mathematically given by the following equation: ρ z z tt � H x − P 2 + F 2 , ρ u u tt � T x + P 1 + F 1 , (1) where the constituents z and u represent the displacement of the fuid and elastic solid material. Te parameters ρ u and ρ z are the densities of each constituent which are assume to be strictly positive constants. T and H are the partial tensions, F 1 and F 2 are the external forces, and P 1 and P 2 are internal body forces associated with the dependent variables u and z. Here, we assume that the constitutive equations of partial tensions are given as in [6] by the following equation: Hindawi Journal of Mathematics Volume 2023, Article ID 6013085, 9 pages https://doi.org/10.1155/2023/6013085