Well-Posedness and Stability Results for Lord Shulman Swelling Porous Thermo-Elastic Soils with Microtemperature and Distributed Delay

: The Lord Shulman swelling porous thermo-elastic soil system with the effects of microtem-perature, temperatures and distributed delay terms is considered in this study. The well-posedness result is established by the Lumer–Phillips corollary applied to the Hille–Yosida theorem. The exponential stability result is proven by the energy method under suitable assumptions.


Introduction and Preliminaries
The first theory that included a viscous liquid, solid and gas mixture was proposed by Eringen [1].The field equations were obtained by investigating this heat-resistant combination [2].The porous media theory, which investigates this type of issue, has also been used to classify expansive (swelling) soils.Due to numerous investigations aimed at mitigating the adverse effects of expansive soils, especially within the fields of architecture and civil engineering, this subject appears promising for further research exploration.For additional information, visit [3][4][5][6][7][8][9].From the linear theory of swelling porous elastic soils, the fundamental field equations are in which ρ ϑ , ρ u > 0 are the densities of the elastic solid material and fluid, while their respective displacements are denoted by u, ϑ.Furthermore, (P 1 , G 1 , H 1 ) represent the partial tension, internal forces of body and eternal forces acting on the displacement.(P 2 , G 2 , H 2 ) are similar, but applied to the elastic solid.Also, the constitutive equations for partial tensions are provided by where a 1 , a 3 > 0, and a 2 = 0 is a real number.A is matrix positive definite with a 1 a 3 > a 2 2 .Quintanilla [9] studied (1) by considering where ξ > 0; the exponential stability can be achieved.Also, in [10], the researchers considered (1) by taking different conditions: where the internal viscous damping function γ(x) has a positive mean.They were able to determine the exponential stability using the spectral approach.To discover more, read [9][10][11][12][13][14][15][16].Time delays are of significant importance in the majority of natural phenomena and industrial systems, as they have the potential to induce instability and should be treated with utmost consideration.Additionally, there are numerous works that have examined this category of issues, including [17][18][19][20][21][22][23].
Numerous researchers worked on similar problems in the literature from different perspectives [24][25][26][27][28].In recent times, there has been a substantial surge of interest among scientists in Lord Shulman's thermo-elasticity, leading to an extensive collection of contributions aimed at elucidating this theory.This theoretical framework encompasses the examination of a system comprising four hyperbolic equations coupled with heat transfer dynamics.Moreover, Lord Shulman thermo-elastic theory was introduced to usher in a more robust heat conduction law, as it concerns thermo-elastic materials exhibiting elastic vibrations.Notably, the heat equation within this context is itself hyperbolic and parallels the equation initially formulated by Fourier's law.To delve deeper into the specifics and gain a comprehensive understanding of this theory, it is recommended that the reader consult the following papers: [29,30].The core evolutionary equations governing onedimensional models of porous thermo-elasticity, incorporating both microtemperature and temperature effects [31][32][33][34], can be expressed as follows: In the context provided, the symbols T, T 0 , H, E, η, q, G, Q and P * denote the stress, reference temperature, equilibrated stress, first energy moment, entropy, heat flux vector, equilibrated body force, mean heat flux and first heat flux moment, respectively.For simplicity in computations, we set T 0 to be equal to 1.
This paper addresses the inherent counterpart of microtemperatures within the Lord Shulman theory.In this scenario, it becomes possible to adapt the constitutive equations in the subsequent manner: in which the microtemperature vector is indicated by , κ > 0 is the relaxation parameter and ρ u , ρ ϑ , a 1 , a 2 , a 3 , β 1 , β 2 > 0. The coefficients γ 0 , k, γ 1 denote the coupling between the temperature and displacement, the thermal conductivity, the coupling between the volume fraction and the temperature, respectively.
Taking a 2 = 0 and the coefficients k 1 , k 2 , k 3 , γ 2 > 0 satisfies the inequalities In the current work, we focus on the thermal effects, which is why we make the assumption β 1 , β 2 > 0 for heat capicity.To add interest to the problem, we also add a distributed delay term to the second equation, creating a new case that differs from earlier research.Under the right assumptions, the system is shown to be well posed, and we use the energy method to demonstrate the result of the exponential stability.
In this investigation, we delve into the realm of the Lord Shulman model for swelling porous thermo-elastic soils, incorporating the influence of microtemperature, temperatures and distributed delay components.Our focus lies in demonstrating the system's wellposedness and examining the outcomes related to its exponential stability.This work is structured as follows: in Section 2, the well-posedness is illustrated, and the exponential stability is demonstrated in Section 3. We state that c > 0 in each of the sentences that follow.

Well-Posedness
Here, we will establish the well-posedness of the system (9)- (11).The following vector function is first introduced: where variables v = u t , ϕ = ϑ t , χ = θ t , Σ = t ; then, the system ( 9) is written as follows: where T : S(T ) ⊂ V :→ V is a linear operator given by in which energy space is denoted by V, such that for any with the following inner product: The domain of T is given by Clearly, S(T ) is dense in V.
Lemma 4. The functional Proof.Direct computations give Estimate (52) easily follows by utilizing Young's inequality.
Lemma 5.The functional Proof.Direct computations give Further simplification of (53) leads us to By differentiating (57) and using (39), (47), (50), ( 52), ( 53) and (55), we have by setting we obtain Now, we choose our constants.We take N 2 large enough, such that then, we pick N 1 large enough, in such a way that Then, we pick N 4 and N 5 large enough, in such a way that Thus, we obtain that  Similarly, by (kk 3 > k 2 1 ), we have