Exponential stabilization of swelling porous systems with thermoelastic damping

Tijani A. Apalara; Moruf O. Yusuf; Soh E. Mukiawa; Ohud B. Almutairi

doi:10.1016/j.jksus.2022.102460

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Exponential stabilization of swelling porous systems with thermoelastic damping

Tijani A. Apalara^{^a,⁎}, Moruf O. Yusuf^{^b}, Soh E. Mukiawa^{^a}, Ohud B. Almutairi^{^a}

a Department of Mathematics, University of Hafr Al- Batin (UHB), Hafr Al- Batin 31991, Saudi Arabia

b Department of Civil Engineering, University of Hafr Al- Batin (UHB), Hafr Al- Batin 31991, Saudi Arabia

⁎Corresponding author. tijani@uhb.edu.sa (Tijani A. Apalara),

Received: 2022-6-9, Accepted: 2022-11-17,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

We study two systems of swelling porous thermoelasticity with minimum damping. Employing the standard multiplier method, we stabilize the systems exponentially without imposing restrictions on the wave velocities of the systems. This result is contrary to those obtained for closely related systems (like Timoshenko and porous systems) with similar single damping. In such scenarios, the authors established that a single damping term is insufficient to exponentially stabilize the system unless the assumption of the equality of wave velocities is imposed.

Keywords

Swelling porous

Thermoelasticity

Exponential stability

93D23

35B40

74F05

1 Introduction

It is imperative to study the characteristics of engineering materials for energy conservation, safety, environmental sustainability, and the effective use of thermal insulation for mechanical and civil engineering applications (Thomas and Rees, 2009). Understanding the thermal properties of materials, such as thermal conductivity and diffusivity, could provide further details about their thermal performance under elastic loading conditions where a restrained strain will induce stresses. Several studies have addressed different methods to test these materials thermal related properties for engineering and energy conservation applications (Low et al., 2013; Sundberg, 1988; Upadhyay et al., 2011 ). It was established in Liu et al. (2020) and López-Acosta et al. (2021) that understanding the thermoelastic properties of the material enhances their effective utilization for foundation stability. Meanwhile, Ghorbani et al. (2017) proved that the presence of moisture could inversely affect the peak velocity under dynamic vibration at elevated temperatures. This could occur to the soil under structural foundations due to earthquakes which could induce more significant shear stress due to thermal changes of underline soil materials. Seasonal changes such as summer and winter could cause changes in the behavior of the substructure soil characteristics due to temperature gradient.

It is crucial to provide solutions towards understanding the complexity of swelling porous using mathematical formulations because it enables scientists and engineers to adequately comprehend the swelling soil’s behavior under several loading and damping conditions. Mathematically, the fundamental system of equations governing the swelling porous thermoelastic soils (see Eringen, 1994; Quintanilla, 2002) is formulated as

(1.1)

\begin{matrix} ρ_{1} u_{tt} - a_{1} u_{xx} - a_{2} v_{xx} - β_{1} Θ_{x} + ξ (u_{xt} - v_{xt}) - α u_{xxt} = 0, \\ ρ_{2} v_{tt} - μ_{1} v_{xx} - a_{2} u_{xx} - β_{2} Θ_{x} - ξ (u_{xt} - v_{xt}) = 0, \\ ρ Θ_{t} - μ_{2} Θ_{xx} - β_{1} u_{xt} - β_{2} v_{xt} = 0, \end{matrix}

where the unknown variables

u, v, Θ : (0, L) \times [0, \infty) \to R

, respectively, symbolize the fluid displacement, the elastic solid material, and the temperature variation. The positive parameters

ρ_{1}, ρ_{2}

, designate the densities relate to z and u, respectively, and

ρ

denotes the heat capacity. The other parameters

ξ, α, a_{i}, μ_{i}, β_{i}, i = 1, 2

are constitute constants satisfying

a_{i} > 0, μ_{i} > 0, β_{i} > 0, ξ > 0, α > 0, a_{1} μ_{1} > a_{2}^{2} .

Quintanilla (2002) considered (1.1) in the isothermal case,

Θ = 0

, which necessitates that

β_{1} = β_{2} = 0

, that is

(1.2)

\begin{matrix} ρ_{1} u_{tt} - a_{1} u_{xx} - a_{2} v_{xx} + ξ (u_{xt} - v_{xt}) - α u_{xxt} = 0, \\ ρ_{2} v_{tt} - μ_{1} v_{xx} - a_{2} u_{xx} - ξ (u_{xt} - v_{xt}) = 0 . \end{matrix}

Employing the standard energy method, he proved that system (1.2) decay exponentially. In addition, using Hurwitz theorem, he showed that system (1.2) with

ξ = 0

also decay exponentially. It is obvious that system (1.1) in its general thermal case is exponentially stable. Further on the case

ξ = 0

, we mention the work of Wang and Guo (2006). They considered (1.2) with

ξ = 0

and replaced the term

- α u_{xxt}

with

α (x) u_{t}

, that is

(1.3)

\begin{matrix} ρ_{1} u_{tt} - a_{1} u_{xx} - a_{2} v_{xx} + α (x) u_{t} = 0, \\ ρ_{2} v_{tt} - μ_{1} v_{xx} - a_{2} u_{xx} = 0, \end{matrix}

and stabilized the system exponentially via the spectral method. Ramos et al. (2020) recently worked on system (1.2) with

ξ = α = 0

and a nonlinear feedback

α (t) g (v_{t})

added to the second equation, that is

(1.4)

\begin{matrix} ρ_{1} u_{tt} - a_{1} u_{xx} - a_{2} v_{xx} = 0, \\ ρ_{2} v_{tt} - μ_{1} v_{xx} - a_{2} u_{xx} + α (t) g (v_{t}) = 0 . \end{matrix}

They obtained an exponential stability result under the assumption of equality of wave velocities, that is

(1.5)

\frac{a_{1}}{ρ_{1}} = \frac{μ_{1}}{ρ_{2}} .

Meanwhile, Apalara et al. (2021a) proved the same exponential stability result for system (1.4) regardless of assumption (1.5). In a similar work, Apalara (2020) investigated system (1.4) replacing the nonlinear feedback with a viscoelastic damping represented by

\int_{0}^{t} g (t - s) v_{xx} (s) ds

, that is,

(1.6)

\begin{matrix} ρ_{1} u_{tt} - a_{1} u_{xx} - a_{2} v_{xx} = 0, \\ ρ_{2} v_{tt} - μ_{1} v_{xx} - a_{2} u_{xx} + \int_{0}^{t} g (t - s) v_{xx} (s) ds = 0, \end{matrix}

and achieved a general stability result without imposing the assumption of equality of the system’s wave velocities. Readers are invited to consult (Feng et al., 2022; Quintanilla, 2004; Quintanilla, 2002; Ramos et al., 2021; Choucha et al., 2021; Ramos et al., 2022; Apalara et al., 2022a; Apalara et al., 2021b; Al-Mahdi et al., 2021; Al-Mahdi et al., 2022a; Al-Mahdi et al., 2022b; Al-Mahdi et al., 2022c; Youkana et al., 2022; Baibeche et al., 2022 ) and the references therein for more fascinating results.

In this article, we consider two independent swelling porous classical thermoelastic systems with minimum damping terms. The first system is

(1.7)

\begin{matrix} ρ_{1} u_{tt} - a_{1} u_{xx} - a_{2} v_{xx} = 0 & in (0, L) \times [0, \infty), \\ ρ_{2} v_{tt} - μ_{1} v_{xx} - a_{2} u_{xx} - β_{2} Θ_{x} = 0 & in (0, L) \times [0, \infty), \\ ρ Θ_{t} - μ_{2} Θ_{xx} - β_{2} v_{xt} = 0 & in (0, L) \times [0, \infty) \end{matrix}

and the second system is

(1.8)

\begin{matrix} ρ_{1} u_{tt} - a_{1} u_{xx} - a_{2} v_{xx} - β_{1} Θ_{x} = 0 & in (0, L) \times [0, \infty), \\ ρ_{2} v_{tt} - μ_{1} v_{xx} - a_{2} u_{xx} = 0 & in (0, L) \times [0, \infty), \\ ρ Θ_{t} - μ_{2} Θ_{xx} - β_{1} u_{xt} = 0 & in (0, L) \times [0, \infty) . \end{matrix}

Each of the systems is supplemented with initial conditions:

(1.9)

u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), v (x, 0) = v_{0} (x), v_{t} (x, 0) = v_{1} (x), Θ (x, 0) = Θ_{0} (x), x \in (0, L)

and boundary conditions:

(1.10)

u (0, t) = u (L, t) = v (0, t) = v (L, t) = Θ_{x} (0, t) = Θ_{x} (L, t) = 0, t \in [0, \infty) .

In each case, we utilize the standard energy method (also known as the multiplier method) to demonstrate exponential stability results notwithstanding the system’s wave velocities. The result is unexpected as opposed to the result obtained for similar systems like thermoelastic porous, where the exponential stability depends on the wave velocities. For example, in Casas and Quintanilla (2005), Casas and Quintanilla considered

(1.11)

\begin{matrix} ρ u_{tt} - μ_{1} u_{xx} - {bv}_{x} + β Θ_{x} = 0, \\ {Jv}_{tt} - α v_{xx} + {bu}_{x} + ξ v - m θ = 0, \\ a Θ_{t} - μ_{2} Θ_{xx} + β u_{xt} + {mv}_{t} = 0 \end{matrix}

and proved a slow decay of solutions. The same result was obtained by Pamplona et al. (2009) when

γ v_{xxt}

was added to the fluid displacement equation in (1.11). However, Santos et al. (2019) extended the slow decay result obtained in Casas and Quintanilla (2005) to exponential stability under the equality of the system’s wave velocities. Analogous result was achieved by Apalara in Apalara (2019). For the thermoelastic Timoshenko system, we mention the result of Rivera and Racke (2002), where they considered

(1.12)

\begin{matrix} ρ_{1} u_{tt} - κ {(u_{x} + v)}_{x} = 0, \\ ρ_{2} v_{tt} - {bv}_{xx} + κ (u_{x} + v) + γ Θ_{x} = 0, \\ ρ Θ_{t} - μ_{2} Θ_{xx} + γ v_{xt} = 0 \end{matrix}

and demonstrated an exponential stability result owing to the assumption

\frac{κ}{ρ_{1}} = \frac{b}{ρ_{2}}

. See (Júnior et al., 2014) for similar result. In all these systems, we see that the exponential stability results depend on the equality of wave velocities of the system. Interestingly, in our systems ((1.7) and (1.8)), we establish the exponential decay results without any restrictions on the wave velocities or any other relationship between the system’s parameters. We must say that the result is similar to the result obtained for the thermoelastic Timoshenko system free of the second spectrum (Apalara et al., 2022b).

Using the boundary conditions (1.10), it follows from the last equation in (1.7)(which is also true for (1.8)) that $\frac{d}{dt} (\int_{0}^{L} Θ (x, t) dx) = 0 .$ So, using the intial condition (1.9), we have $\int_{0}^{L} Θ (x, t) dx = \int_{0}^{L} Θ_{0} (x) dx$ . By setting $θ (x, t) = Θ (x, t) - \frac{1}{L} \int_{0}^{L} Θ_{0} (x) dx$ we get $\int_{0}^{L} θ (x, t) dx = 0$ . Consequently, the usage of Poincaré’s inequality is appropriate to the Neumann boundary conditions on $Θ$ . Furthermore, it is obvious that $θ_{t} = Θ_{t}, θ_{x} = Θ_{x}$ , and $θ_{xx} = Θ_{xx}$ . Accordingly, systems (1.7), (1.9)–(1.10) and (1.8), (1.9)–(1.10), respectively, transform to

(1.13)

\begin{matrix} ρ_{1} u_{tt} - a_{1} u_{xx} - a_{2} v_{xx} = 0, \\ ρ_{2} v_{tt} - μ_{1} v_{xx} - a_{2} u_{xx} - β_{2} θ_{x} = 0, \\ ρ θ_{t} - μ_{2} θ_{xx} - β_{2} v_{xt} = 0, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), v (x, 0) = v_{0} (x), \\ v_{t} (x, 0) = v_{1} (x), θ (x, 0) = Θ_{0} (x) - \frac{1}{L} \int_{0}^{L} Θ_{0} (x) dx, \\ u (0, t) = u (L, t) = v (0, t) = v (L, t) = θ_{x} (0, t) = θ_{x} (L, t) = 0 \end{matrix}

and

(1.14)

\begin{matrix} ρ_{1} u_{tt} - a_{1} u_{xx} - a_{2} v_{xx} - β_{1} θ_{x} = 0, \\ ρ_{2} v_{tt} - μ_{1} v_{xx} - a_{2} u_{xx} = 0, \\ ρ θ_{t} - μ_{2} θ_{xx} - β_{1} u_{xt} = 0, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), v (x, 0) = v_{0} (x), \\ v_{t} (x, 0) = v_{1} (x), θ (x, 0) = Θ_{0} (x) - \frac{1}{L} \int_{0}^{L} Θ_{0} (x) dx, \\ u (0, t) = u (L, t) = v (0, t) = v (L, t) = θ_{x} (0, t) = θ_{x} (L, t) = 0, \end{matrix}

for

(x, t) \in (0, L) \times [0, \infty)

. Moving forward, we consider systems (1.13) and (1.14). The following is the outline of the remaining sections: In Section 2, we demonstrate an exponential stability result for (1.13). Section 3 deals with the proof of an exponential stability result for (1.14). We end the paper with conclusions in Section 4. Throughout this article, the letter

c_{p}

in the estimates denotes a Poincaré’s constant.

2 Stability result for (1.7), (1.13)

Specific to our system, we have the following suitable Lyapunov functional

(2.1)

P (t) ≔ C E (t) + \sum_{i = 1}^{4} C_{i} F_{i} (t), \forall t \geq 0,

where

C > 0, C_{i} > 0, i = 1 \dots 4

, are constants to be discreetly chosen later.

F_{i}, i = 1 \dots 4

, are auxiliary functionals given by

(2.2)

F_{1} (t) ≔ a_{2} ρ_{2} \int_{0}^{L} (v_{t} u - u_{t} v) dx,

(2.3)

F_{2} (t) ≔ ρ_{2} \int_{0}^{L} v_{t} vdx - \frac{a_{2}}{a_{1}} ρ_{1} \int_{0}^{L} u_{t} vdx,

(2.4)

F_{3} (t) ≔ - ρ \int_{0}^{L} v_{t} \int_{0}^{x} θ (y) dydx,

(2.5)

F_{4} (t) ≔ - \int_{0}^{L} u_{t} udx,

and E represents the energy of the system (it is the same for both systems) defined by

(2.6)

E (t) = \frac{1}{2} \int_{0}^{L} [ρ_{1} u_{t}^{2} + a_{1} u_{x}^{2} + ρ_{2} v_{t}^{2} + μ_{1} v_{x}^{2} + ρ θ^{2} + 2 a_{2} u_{x} v_{x}] dx, \forall t \geq 0 .

The routine implementation of both Young’s inequality as well as Cauchy–Schwarz inequality leads to the critical fact that the Lyapunov functional

P

is equivalent to the energy E provided that

C

is sufficiently large. Specifically, for large

C

and some constants

ℓ_{1}, ℓ_{2} > 0

, we have

(2.7)

ℓ_{1} E (t) ⩽ P (t) ⩽ ℓ_{2} E (t), \forall t \geq 0 .

The next five lemmas capture the derivatives of the energy functional E and the estimate of the derivatives of functionals

F_{i}, i = 1 \dots 4

. Afterwards, we state and prove Theorem 2.6 which deals with the exponential stability result for system (1.13).

Lemma 2.1

The energy E of system (1.13) defined by (2.6), satisfies

(2.8)

E^{'} (t) = - μ_{2} \int_{0}^{L} θ_{x}^{2} dx \leq 0, \forall t \geq 0 .

Proof

By respectively multiplying (1.13)₁, (1.13)₂, and (1.13)₃ by $u_{t}, v_{t}$ , and $θ$ , then using integration by parts with respect to x on the product, we get

(2.9)

\frac{ρ_{1}}{2} \frac{d}{dt} \int_{0}^{L} u_{t}^{2} dx + \frac{a_{1}}{2} \frac{d}{dt} \int_{0}^{L} u_{x}^{2} dx + a_{2} \frac{d}{dt} \int_{0}^{L} u_{x} v_{x} dx - a_{2} \int_{0}^{L} u_{x} v_{xt} dx = 0, \forall t \geq 0,

(2.10)

\frac{ρ_{2}}{2} \frac{d}{dt} \int_{0}^{L} v_{t}^{2} dx + \frac{μ_{1}}{2} \frac{d}{dt} \int_{0}^{L} v_{x}^{2} dx + a_{2} \int_{0}^{L} u_{x} v_{xt} dx + β_{2} \int_{0}^{L} v_{xt} θ dx = 0, \forall t \geq 0,

(2.11)

\frac{ρ}{2} \frac{d}{dt} \int_{0}^{L} θ^{2} dx + μ_{2} \int_{0}^{L} θ_{x}^{2} dx - β_{2} \int_{0}^{L} v_{xt} θ dx = 0, \forall t \geq 0 .

Summing (2.9)–(2.11) and considering (2.6), yields (2.8). square

Lemma 2.2

The functional $F_{1}$ , defined by (2.2), satisfies

(2.12)

\begin{matrix} F_{1}^{'} (t) ⩽ - & \frac{a_{2}^{2}}{2} \int_{0}^{L} u_{x}^{2} dx + (\frac{a_{2}^{2} ρ_{2}}{ρ_{1}} + {(\frac{a_{1} ρ_{2}}{ρ_{1}} - μ_{1})}^{2}) \int_{0}^{L} v_{x}^{2} dx + β_{2}^{2} c_{p} \int_{0}^{L} θ_{x}^{2} dx, \forall t \geq 0 . \end{matrix}

Proof

Taking the derivative of (2.2) and using (1.13), we see that, for all $t \geq 0$ ,

(2.13)

F_{1}^{'} (t) = - a_{2}^{2} \int_{0}^{L} u_{x}^{2} dx + \frac{a_{2}^{2} ρ_{2}}{ρ_{1}} \int_{0}^{L} v_{x}^{2} dx + a_{2} (\frac{a_{1} ρ_{2}}{ρ_{1}} - μ_{1}) \int_{0}^{L} u_{x} v_{x} dx - a_{2} β_{2} \int_{0}^{L} u_{x} θ dx .

By first utilizing Young’s inequality followed by Poincaré’s inequality, we get,

(2.14)

a_{2} (\frac{a_{1} ρ_{2}}{ρ_{1}} - μ_{1}) \int_{0}^{L} u_{x} v_{x} dx ⩽ \frac{a_{2}^{2}}{4} \int_{0}^{L} u_{x}^{2} dx + {(\frac{a_{1} ρ_{2}}{ρ_{1}} - μ)}^{2} \int_{0}^{L} v_{x}^{2} dx, \forall t \geq 0,

(2.15)

- a_{2} β_{2} \int_{0}^{L} u_{x} θ dx ⩽ \frac{a_{2}^{2}}{4} \int_{0}^{L} u_{x}^{2} dx + β_{2}^{2} c_{p} \int_{0}^{L} θ_{x}^{2} dx, \forall t \geq 0 .

The substitution of estimates (2.14) and (2.15) into (2.13) gives (2.12). square

Lemma 2.3

The functional $F_{2}$ , defined by (2.3), satisfies, for any $ε_{1} > 0$ and some constant $m_{0} > 0$ ,

(2.16)

F_{2}^{'} (t) ⩽ - \frac{m_{0}}{2} \int_{0}^{L} v_{x}^{2} dx + ε_{1} \int_{0}^{L} u_{t}^{2} dx + (ρ_{2} + \frac{a_{2}^{2} ρ_{1}^{2}}{4 a_{1}^{2} ε_{1}}) \int_{0}^{L} v_{t}^{2} dx + \frac{β_{2}^{2} c_{p}}{2 m_{0}} \int_{0}^{L} θ_{x}^{2} dx, \forall t \geq 0 .

Proof

Differentiating $F_{2}$ and adapting (1.13)₁ and (1.13)₂ for terms $u_{tt}$ and $v_{tt}$ , respectively, we get

(2.17)

F_{2}^{'} (t) = - (μ_{1} - \frac{a_{2}^{2}}{a_{1}}) \int_{0}^{L} v_{x}^{2} dx + ρ_{2} \int_{0}^{L} v_{t}^{2} dx - \frac{a_{2} ρ_{1}}{a_{1}} \int_{0}^{L} u_{t} v_{t} dx - β_{2} \int_{0}^{L} v_{x} θ dx, \forall t \geq 0 .

Implementing Young’s inequality for any

σ_{1} > 0, ε_{1} > 0

, followed by Poincaré’s inequality, we achieve

(2.18)

- \frac{a_{2} ρ_{1}}{a_{1}} \int_{0}^{L} u_{t} v_{t} dx ⩽ ε_{1} \int_{0}^{L} u_{t}^{2} dx + \frac{a_{2}^{2} ρ_{1}^{2}}{4 a_{1}^{2} ε_{1}} \int_{0}^{L} v_{t}^{2} dx,

(2.19)

- β_{2} \int_{0}^{L} v_{x} θ dx ⩽ σ_{1} \int_{0}^{L} v_{x}^{2} dx + \frac{β_{2}^{2} c_{p}}{4 σ_{1}} \int_{0}^{L} θ_{x}^{2} dx .

Direct substitution of (2.18) and (2.19) into (2.17) gives

F_{2}^{'} (t) ⩽ - (μ_{1} - \frac{a_{2}^{2}}{a_{1}} - σ_{1}) \int_{0}^{L} v_{x}^{2} dx + ε_{1} \int_{0}^{L} u_{t}^{2} dx + (ρ_{2} + \frac{a_{2}^{2} ρ_{1}^{2}}{4 a_{1}^{2} ε_{1}}) \int_{0}^{L} v_{t}^{2} dx + \frac{β_{2}^{2} c_{p}}{4 σ_{1}} \int_{0}^{L} θ_{x}^{2} dx, \forall t \geq 0 .

Using the fact that

a_{1} μ_{1} > a_{2}^{2}

, we have

m_{0} = μ_{1} - \frac{a_{2}^{2}}{a_{1}} > 0

. So, by taking

σ_{1} = \frac{m_{0}}{2}

, we end up with estimate (2.16). square

Lemma 2.4

The functional $F_{3}$ , defined by (2.4), satisfies, for any $ε_{2}, ε_{3} > 0$ , the estimate

(2.20)

\begin{matrix} F_{3}^{'} (t) ⩽ - & \frac{β_{2}}{2} \int_{0}^{L} v_{t}^{2} dx + ε_{2} \int_{0}^{L} u_{x}^{2} dx + ε_{3} \int_{0}^{L} v_{x}^{2} dx + (\frac{ρ β_{2} c_{p}}{ρ_{2}} + \frac{ρ^{2} a_{2}^{2} c_{p}}{4 ε_{2} ρ_{2}^{2}} + \frac{ρ^{2} μ_{1}^{2} c_{p}}{4 ε_{3} ρ_{2}^{2}} + \frac{μ_{2}^{2}}{2 β_{2}}) \int_{0}^{L} θ_{x}^{2} dx . \end{matrix}

Proof

Employing (1.13), we obtain, for all $t \geq 0$ ,

(2.21)

\begin{matrix} F_{3}^{'} (t) = - & β_{2} \int_{0}^{L} v_{t}^{2} dx + \frac{ρ β_{2}}{ρ_{2}} \int_{0}^{L} θ^{2} dx + \frac{ρ a_{2}}{ρ_{2}} \int_{0}^{L} u_{x} θ dx + \frac{ρ μ_{1}}{ρ_{2}} \int_{0}^{L} v_{x} θ dx - μ_{2} \int_{0}^{L} v_{t} θ_{x} dx . \end{matrix}

Similar to the proof of the previous two lemmas, by enforcing Young’s inequality for any

ε_{2}, ε_{3} > 0

, and Poincaré’s inequality, the last three integrals of (2.21), give

(2.22)

\frac{ρ a_{2}}{ρ_{2}} \int_{0}^{L} u_{x} θ dx ⩽ ε_{2} \int_{0}^{L} u_{x}^{2} dx + \frac{ρ^{2} a_{2}^{2} c_{p}}{4 ε_{2} ρ_{2}^{2}} \int_{0}^{L} θ_{x}^{2} dx,

(2.23)

\frac{ρ μ_{1}}{ρ_{2}} \int_{0}^{L} v_{x} θ dx ⩽ ε_{3} \int_{0}^{L} v_{x}^{2} dx + \frac{ρ^{2} μ_{1}^{2} c_{p}}{4 ε_{3} ρ_{2}^{2}} \int_{0}^{L} θ_{x}^{2} dx,

(2.24)

- μ_{2} \int_{0}^{L} v_{t} θ_{x} dx ⩽ \frac{β_{2}}{2} \int_{0}^{L} v_{t}^{2} dx + \frac{μ_{2}^{2}}{2 β_{2}} \int_{0}^{L} θ_{x}^{2} dx .

The combination of (2.21)–(2.24) gives (2.20). square

Lemma 2.5

The functional $F_{4}$ , defined by (2.5), satisfies

(2.25)

F_{4}^{'} (t) ⩽ - \int_{0}^{L} u_{t}^{2} dx + \frac{2 a_{1}}{ρ_{1}} \int_{0}^{L} u_{x}^{2} dx + \frac{μ_{1}}{4 ρ_{1}} \int_{0}^{L} v_{x}^{2} dx, \forall t \geq 0 .

Proof

Using (1.13), it is clear that $F_{4}^{'} (t) = - \int_{0}^{L} u_{t}^{2} dx + \frac{a_{1}}{ρ_{1}} \int_{0}^{L} u_{x}^{2} dx + \frac{a_{2}}{ρ_{1}} \int_{0}^{L} u_{x} v_{x} dx, \forall t \geq 0 .$ Using Young’s inequality, we get $\begin{matrix} \frac{a_{2}}{ρ_{1}} \int_{0}^{L} u_{x} v_{x} dx ⩽ \frac{a_{1}}{ρ_{1}} \int_{0}^{L} u_{x}^{2} dx + \frac{a_{2}^{2}}{4 a_{1} ρ_{1}} \int_{0}^{L} v_{x}^{2} dx \\ ⩽ \frac{a_{1}}{ρ_{1}} \int_{0}^{L} u_{x}^{2} dx + \frac{μ_{1}}{4 ρ_{1}} \int_{0}^{L} v_{x}^{2} dx using a_{1} μ_{1} > a_{2}^{2} . \end{matrix}$ Consequently, we end up with (2.25). square

Having achieved the necessary estimates, we now focus on the theorem which captures our first result.

Theorem 2.6

The problem (1.13) is exponentially stable, that is, for some constants $k_{0}, k_{1} > 0$ , the energy of system (1.13) defined by (2.6) satisfies

(2.26)

E (t) ⩽ k_{0} e^{- k_{1} t} \forall t \geq 0 .

Proof

Recall the Lyapunov functional defined by (2.1), that is $P (t) ≔ C E (t) + \sum_{i = 1}^{4} C_{i} F_{i} (t), \forall t \geq 0,$ and using the estimates (2.8), (2.12), (2.16)–(2.25), together with setting $C_{4} = 2, ε_{1} = \frac{1}{C_{2}}, ε_{2} = \frac{a_{2}^{2} C_{1}}{4 C_{3}} ε_{3} = \frac{m_{0} C_{2}}{4 C_{3}},$ we conclude that $\begin{matrix} P^{'} (t) ⩽ - \int_{0}^{L} u_{t}^{2} dx - [\frac{m_{0}}{4} C_{2} - (\frac{a_{2}^{2} ρ_{2}}{ρ_{1}} + {(\frac{a_{1} ρ_{2}}{ρ_{1}} - μ_{1})}^{2}) C_{1} - \frac{μ_{1}}{2 ρ_{1}}] \int_{0}^{L} v_{x}^{2} dx \\ - [μ_{2} C - β_{2}^{2} c_{p} C_{1} - \frac{β_{2}^{2} c_{p}}{2 m_{0}} C_{2} - (\frac{μ_{2}^{2}}{2 β_{2}} + \frac{ρ β_{2} c_{p}}{ρ_{2}} + \frac{ρ^{2} c_{p} C_{3}}{ρ_{2}^{2} C_{1}} + \frac{ρ^{2} μ_{1}^{2} c_{p} C_{3}}{m_{0} ρ_{2}^{2} C_{2}}) C_{3}] \int_{0}^{L} θ_{x}^{2} dx \\ - [\frac{a_{2}^{2}}{4} C_{1} - \frac{4 a_{1}}{ρ_{1}}] \int_{0}^{L} u_{x}^{2} dx - [\frac{β_{2}}{2} C_{3} - (ρ_{2} + \frac{a_{2}^{2} ρ_{1}^{2} C_{2}}{4 a_{1}^{2}}) C_{2}] \int_{0}^{L} v_{t}^{2} dx, \forall t \geq 0 . \end{matrix}$ By consecutively letting $\{\begin{matrix} C_{1} = 4 a_{2}^{- 2} (\frac{4 a_{1}}{ρ_{1}} + 1), \\ C_{2} = 4 m_{0}^{- 1} ((\frac{a_{2}^{2} ρ_{2}}{ρ_{1}} + {(\frac{a_{1} ρ_{2}}{ρ_{1}} - μ_{1})}^{2}) C_{1} + \frac{μ_{1}}{2 ρ_{1}} + 1), \\ C_{3} = 2 β_{2}^{- 1} ((ρ_{2} + \frac{a_{2}^{2} ρ_{1}^{2} C_{2}}{4 a_{1}^{2}}) C_{2} + 1), \\ C = μ_{2}^{- 1} (β_{2}^{2} c_{p} C_{1} + \frac{β_{2}^{2} c_{p}}{2 m_{0}} C_{2} + (\frac{μ_{2}^{2}}{2 β_{2}} + \frac{ρ β_{2} c_{p}}{ρ_{2}} + \frac{ρ^{2} c_{p} C_{3}}{ρ_{2}^{2} C_{1}} + \frac{ρ^{2} μ_{1}^{2} c_{p} C_{3}}{m_{0} ρ_{2}^{2} C_{2}}) C_{3} + c_{p}) \end{matrix}$ and using Poincaré’s inequality $- c_{p} \int_{0}^{L} θ_{x}^{2} dx ⩽ - \int_{0}^{L} θ^{2} dx, \forall t \geq 0,$ it turns out that

(2.27)

\begin{matrix} P^{'} (t) ⩽ & - \int_{0}^{L} (u_{t}^{2} + u_{x}^{2} + v_{t}^{2} + v_{x}^{2} + θ^{2}) dx, \forall t \geq 0 . \end{matrix}

Meanwhile, by recalling the energy functional defined by (2.6)

E (t) = \frac{1}{2} \int_{0}^{L} [ρ_{1} u_{t}^{2} + a_{1} u_{x}^{2} + ρ_{2} v_{t}^{2} + μ_{1} v_{x}^{2} + ρ θ^{2} + 2 a_{2} u_{x} v_{x}] dx, \forall t \geq 0,

and using Young’s inequality, we have, for some constant

c > 0

E (t) ⩽ c \int_{0}^{L} [u_{t}^{2} + u_{x}^{2} + v_{t}^{2} + v_{x}^{2} + θ^{2}] dx, \forall t > 0

which implies

(2.28)

- \int_{0}^{L} [u_{t}^{2} + u_{x}^{2} + v_{t}^{2} + v_{x}^{2} + θ^{2}] dx ⩽ - c_{1} E (t), \forall t > 0,

where

c_{1} = \frac{1}{c} > 0

. Consequently, by merging (2.27) and (2.28), we have

P^{'} (t) ⩽ - c_{1} E (t) .

Using (2.7), we get

P^{'} (t) ⩽ - \frac{c_{1}}{ℓ_{2}} P (t) .

Solving the differential inequality, we arrive at

P (t) ⩽ P (0) \exp (- \frac{c_{1}}{ℓ_{2}}) .

Finally, using (2.7) once again, we obtain (2.26), with

k_{0} = \frac{ℓ_{2}}{ℓ_{1}} E (0) > 0

and

k_{1} = \frac{c_{1}}{ℓ_{2}} > 0

. Thus, we conclude the proof of Theorem 2.6. square

3 Stability result for (1.14)

Similar to Section 2, we specify the following Lyapunov functional

(3.1)

Q (t) ≔ M E (t) + \sum_{i = 1}^{4} M_{i} G_{i} (t), \forall t \geq 0,

where

M, M_{i}, i = 1 \dots 4

, are positive constants to be carefully selected subsequently.

G_{i}, i = 1 \dots 4

, are auxiliary functionals given by, for all

t \geq 0

(3.2)

G_{1} (t) ≔ a_{2} ρ_{1} \int_{0}^{L} (u_{t} v - v_{t} u) dx,

(3.3)

G_{2} (t) ≔ ρ_{1} \int_{0}^{L} u_{t} udx - \frac{a_{2}}{μ_{1}} ρ_{2} \int_{0}^{L} v_{t} udx,

(3.4)

G_{3} (t) ≔ - ρ \int_{0}^{L} u_{t} \int_{0}^{x} θ (y) dydx,

(3.5)

G_{4} (t) ≔ - \int_{0}^{L} v_{t} vdx

and E remains the same as in Section 2. Using (1.14), the derivative of the functionals satisfy, for all

t \geq 0

(3.6)

G_{1}^{'} (t) ⩽ - \frac{a_{2}^{2}}{2} \int_{0}^{L} v_{x}^{2} dx + k \int_{0}^{L} u_{x}^{2} dx + {kc}_{p} \int_{0}^{L} θ_{x}^{2} dx,

(3.7)

G_{2}^{'} (t) ⩽ - \frac{n_{0}}{2} \int_{0}^{L} u_{x}^{2} dx + ∊_{1} \int_{0}^{L} v_{t}^{2} dx + k (1 + \frac{1}{∊_{1}}) \int_{0}^{L} u_{t}^{2} dx + k \int_{0}^{L} θ_{x}^{2} dx,

(3.8)

G_{3}^{'} (t) ⩽ - \frac{β_{1}}{2} \int_{0}^{L} u_{t}^{2} dx + ∊_{2} \int_{0}^{L} u_{x}^{2} dx + ∊_{3} \int_{0}^{L} v_{x}^{2} dx + k (1 + \frac{1}{∊_{2}} + \frac{1}{∊_{3}}) \int_{0}^{L} θ_{x}^{2} dx,

(3.9)

G_{4}^{'} (t) ⩽ - \int_{0}^{L} v_{t}^{2} dx + k \int_{0}^{L} v_{x}^{2} dx + k \int_{0}^{L} u_{x}^{2} dx,

where k represents a generic positive constant. Since

a_{1} μ_{1} > a_{2}^{2}

, we have

n_{0} = a_{1} - \frac{a_{2}^{2}}{μ_{1}} > 0

. The main thorem of this section is:

Theorem 3.1

The problem (1.14) is exponentially stable, that is, for some constants $κ_{0}, κ_{1} > 0$ , the energy of system (1.14) defined by (2.6) satisfies

(3.10)

E (t) ⩽ κ_{0} e^{- κ_{1} t} \forall t \geq 0 .

Proof

Using the Lyapunov functional defined by (3.1), the estimates (2.8), (3.6)–(3.9), and setting $M_{4} = 2, ∊_{1} = \frac{1}{M_{2}}, ∊_{2} = \frac{n_{0} M_{2}}{4 M_{3}}, ∊_{3} = \frac{a_{2}^{2} M_{1}}{4 M_{3}},$ we end up with $\begin{matrix} Q^{'} (t) ⩽ - [\frac{β_{1}}{2} M_{3} - c M_{2} (1 + M_{2})] \int_{0}^{L} u_{t}^{2} dx - [\frac{n_{0}}{4} M_{2} - c M_{1} - c] \int_{0}^{L} u_{x}^{2} dx - \int_{0}^{L} v_{t}^{2} dx \\ - [\frac{a_{2}^{2}}{4} M_{1} - c] \int_{0}^{L} v_{x}^{2} dx - [k M - c M_{1} - c M_{2} - c M_{3} (1 + \frac{M_{1}}{M_{3}} + \frac{M_{2}}{M_{3}})] \int_{0}^{L} θ_{x}^{2} dx . \end{matrix}$ Emulating the remaining steps in the proof of Theorem 2.6 yields (3.10). This marks the conclusion of our results. square

4 Conclusions

In this paper, we use the energy (also known as multiplier) method to achieve exponential decay results for two independent swelling porous thermoelastic systems where the heat conduction is controlled by the classical Fourier law. The results are obtained without imposing the well-known restrictions on the wave velocities or any other relationship between the coefficients of the system. So, our present results substantially contribute to the theory related to the asymptotic behaviors of swelling porous elastic media and improve earlier endeavors in the literature.

Availability of Data

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Funding

The Deanship of Scientific Research, University of Hafr Al-Batin (UHB), funds this research under project No: 0046–1443-S.

Authors’ contributions

All authors contributed equally. All authors read and approved the final manuscript.

Acknowledgement

The authors extend their appreciation to the Deanship of Scientific Research, University of Hafr Al-Batin (UHB), for funding this work through the research group project No: 0046-1443-S.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

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Apalara T.A., . General stability result of swelling porous elastic soils with a viscoelastic damping. Zeitschrift für angewandte Mathematik und Physik. 2020;71(6):1-10.
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Baibeche S., Bouzettouta L., Guesmia A., Abdelli M., . Well-posedness and exponential stability of swelling porous elastic soils with a second sound and distributed delay term. J. Math. Comput. Sci.. 2022;12 pp. Article-ID
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Quintanilla R., . Exponential stability for one-dimensional problem of swelling porous elastic soils with fluid saturation. J. Comput. Appl. Math.. 2002;145(2):525-533.
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Quintanilla R., . Exponential stability of solutions of swelling porous elastic soils. Meccanica. 2004;39(2):139-145.
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Ramos A., Almeida Júnior D., Freitas M., Noé A., Santos M.D., . Stabilization of swelling porous elastic soils with fluid saturation and delay time terms. J. Mathe. Phys.. 2021;62(2):021507.
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Sundberg, J., 1988. Thermal properties of soils and rocks.
Thomas H.R., Rees S.W., . Measured and simulated heat transfer to foundation soils. Géotechnique. 2009;59(4):365-375.
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Upadhyay S., Chandra H., Joshi M., Joshi D.P., . Thermoelastic properties of minerals at high temperature. Pramana. 2011;76(1):183-188.
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Wang J.-M., Guo B.-Z., . On the stability of swelling porous elastic soils with fluid saturation by one internal damping. IMA J. Appl. Mathe.. 2006;71(4):565-582.
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Youkana A., Al-Mahdi A.M., Messaoudi S.A., . General energy decay result for a viscoelastic swelling porous-elastic system. Zeitschrift für angewandte Mathematik und Physik. 2022;73(3):1-17.
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Introduction

Stability result for (1.7), (1.13)

Stability result for (1.14)

Conclusions

Availability of Data
Funding
Authors’ contributions

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[1] Al-Mahdi A.M., Al-Gharabli M.M., Alahyane M., . Theoretical and numerical stability results for a viscoelastic swelling porous-elastic system with past history. AIMS Mathe.. 2021;6(11):11921-11949.
[Google Scholar]

[2] Al-Mahdi A.M., Messaoudi S.A., Al-Gharabli M.M., . A stability result for a swelling porous system with nonlinear boundary dampings. J. Function Spaces. 2022;22
[Google Scholar]

[3] Al-Mahdi A.M., Al-Gharabli M.M., Alahyane M., . Theoretical and computational results of a memory-type swelling porous-elastic system. Mathe. Comput. Appl.. 2022;27(2):27.
[Google Scholar]

[4] Al-Mahdi A.M., Al-Gharabli M.M., Apalara T.A., . On the stability result of swelling porous-elastic soils with infinite memory. Applicable Anal. 2022:1-17.
[Google Scholar]

[5] Apalara T.A., . On the stability of porous-elastic system with microtemparatures. J. Therm. Stresses. 2019;42(2):265-278.
[Google Scholar]

[6] Apalara T.A., . General stability result of swelling porous elastic soils with a viscoelastic damping. Zeitschrift für angewandte Mathematik und Physik. 2020;71(6):1-10.
[Google Scholar]

[7] Apalara T., Soufyane A., Afilal M., Alahyane M., . A general stability result for swelling porous elastic media with nonlinear damping. In: Applicable Anal.. 2021. p. :1-16.
[Google Scholar]

[8] Apalara T.A., Yusuf M.O., Salami B.A., . On the control of viscoelastic damped swelling porous elastic soils with internal delay feedbacks. J. Mathe. Anal. Appl.. 2021;504(2):125429.
[Google Scholar]

[9] Apalara T.A., Soufyane A., Afilal M., . On well-posedness and exponential decay of swelling porous thermoelastic media with second sound. In: J. Mathe. Anal. Appl.. 2022. p. :126006.
[Google Scholar]

[10] Apalara T.A., Raposo C.A., Ige A., . Thermoelastic timoshenko system free of second spectrum. Appl. Mathe. Lett.. 2022;126:107793.
[Google Scholar]

[11] Baibeche S., Bouzettouta L., Guesmia A., Abdelli M., . Well-posedness and exponential stability of swelling porous elastic soils with a second sound and distributed delay term. J. Math. Comput. Sci.. 2022;12 pp. Article-ID
[Google Scholar]

[12] Casas P.S., Quintanilla R., . Exponential stability in thermoelasticity with microtemperatures. Int. J. Eng. Sci.. 2005;43(1–2):33-47.
[Google Scholar]

[13] Choucha A., Boulaaras S.M., Ouchenane D., Cherif B.B., Abdalla M., . Exponential stability of swelling porous elastic with a viscoelastic damping and distributed delay term. J. Function Spaces. 2021;2021
[Google Scholar]

[14] Eringen A.C., . A continuum theory of swelling porous elastic soils. Int. J. Eng. Sci.. 1994;32(8):1337-1349.
[Google Scholar]

[15] Feng, B., Ramos, A., Júnior, D., Freitas, M., Barbosa, R., 2022. A new stability result for swelling porous elastic media with structural damping. ANNALI DELL’UNIVERSITA’DI FERRARA, pp. 1–14.

[16] Ghorbani, J., Nazem, M., Carter, J., Airey, D., 2017. A numerical study of the effect of moisture content on induced ground vibration during dynamic compaction. In: International Conference on Performance-based Design in Earthquake Geotechnical Engineering 2017, International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE).

[17] Júnior D.d.S.A., Santos M., Rivera J.M., . Stability to 1-d thermoelastic timoshenko beam acting on shear force. Zeitschrift für angewandte Mathematik und Physik. 2014;65(6):1233-1249.
[Google Scholar]

[18] Liu C., Hu X., Yao R., Han Y., Wang Y., He W., Fan H., Du L., . Assessment of soil thermal conductivity based on bpnn optimized by genetic algorithm. Adv. Civil Eng.. 2020;2020
[Google Scholar]

[19] López-Acosta N.P., Zaragoza-Cardiel A.I., Barba-Galdámez D.F., . Determination of thermal conductivity properties of coastal soils for gshps and energy geostructure applications in mexico. Energies. 2021;14(17):5479.
[Google Scholar]

[20] Low J., Loveridge F., Powrie W., . Measuring soil thermal properties for use in energy foundation design. In: Proceedings of the 18th International Conference on Soil Mechanics and Geotechnical Engineering, Paris, France. 2013. p. :2-6.
[Google Scholar]

[21] Pamplona P.X., Rivera J.E.M., Quintanilla R., . Stabilization in elastic solids with voids. J. Mathe. Anal. Appl.. 2009;350(1):37-49.
[Google Scholar]

[22] Quintanilla R., . On the linear problem of swelling porous elastic soils with incompressible fluid. Int. J. Eng. Sci.. 2002;40(13):1485-1494.
[Google Scholar]

[23] Quintanilla R., . Exponential stability for one-dimensional problem of swelling porous elastic soils with fluid saturation. J. Comput. Appl. Math.. 2002;145(2):525-533.
[Google Scholar]

[24] Quintanilla R., . Exponential stability of solutions of swelling porous elastic soils. Meccanica. 2004;39(2):139-145.
[Google Scholar]

[25] Ramos A., Freitas M., Almeida D. Jr, Noé A., Santos M.D., . Stability results for elastic porous media swelling with nonlinear damping. J. Mathe. Phys.. 2020;61((10)101505):1-10.
[Google Scholar]

[26] Ramos A., Almeida Júnior D., Freitas M., Noé A., Santos M.D., . Stabilization of swelling porous elastic soils with fluid saturation and delay time terms. J. Mathe. Phys.. 2021;62(2):021507.
[Google Scholar]

[27] Ramos A., Apalara T., Freitas M., Araújo M., . Equivalence between exponential stabilization and boundary observability for swelling problem. J. Mathe. Phys.. 2022;63(1):011511.
[Google Scholar]

[28] Rivera J.E.M., Racke R., . Mildly dissipative nonlinear timoshenko systems—global existence and exponential stability. J. Mathe. Anal. Appl.. 2002;276(1):248-278.
[Google Scholar]

[29] Santos M., Campelo A., Oliveira M., . On porous-elastic systems with fourier law. Applicable Anal.. 2019;98(6):1181-1197.
[Google Scholar]

[30] Sundberg, J., 1988. Thermal properties of soils and rocks.

[31] Thomas H.R., Rees S.W., . Measured and simulated heat transfer to foundation soils. Géotechnique. 2009;59(4):365-375.
[Google Scholar]

[32] Upadhyay S., Chandra H., Joshi M., Joshi D.P., . Thermoelastic properties of minerals at high temperature. Pramana. 2011;76(1):183-188.
[Google Scholar]

[33] Wang J.-M., Guo B.-Z., . On the stability of swelling porous elastic soils with fluid saturation by one internal damping. IMA J. Appl. Mathe.. 2006;71(4):565-582.
[Google Scholar]

[34] Youkana A., Al-Mahdi A.M., Messaoudi S.A., . General energy decay result for a viscoelastic swelling porous-elastic system. Zeitschrift für angewandte Mathematik und Physik. 2022;73(3):1-17.
[Google Scholar]

Exponential stabilization of swelling porous systems with thermoelastic damping

Abstract

Keywords

Swelling porous

Thermoelasticity

Exponential stability

93D23

35B40

74F05

1 Introduction

2 Stability result for (1.7), (1.13)

3 Stability result for (1.14)

4 Conclusions

Availability of Data

Funding

Authors’ contributions

Acknowledgement

Declaration of Competing Interest

References

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