Existence and stability results of nonlinear swelling equations with logarithmic source terms

1.   Introduction

Swelling soils are a significant environmental issue that has garnered considerable attention from many researchers due to their potential to cause structural damage or destruction. These soils show a tendency to swell in volume when exposed to moisture, primarily due to the presence of clay minerals that naturally attract and absorb water molecules. Upon introducing water to swelling soils, the molecules are drawn into gaps between the soil plates. As the amount of absorbed water increases, the plates are forced further apart, leading to an increase in soil pore pressure. Consequently, swelling soils pose substantial geotechnical and structural challenges to the environment and society. Swelling soils are prevalent worldwide, and recent estimates from the American Society of Civil Engineers suggests that one in four homes experience some form of damage caused by swelling soils. Typically, the financial losses incurred by property owners due to these soils exceed those caused by earthquakes, floods, hurricanes, and tornadoes combined. Therefore, it is important to explore practical methods for eliminating or minimizing the damages caused by swelling soils. Therefore, studying of the asymptotic behavior of swelling porous elastic soils is important for architecture and civil engineering. For more information in the continuum theory of material, we refer the reader to ^[1], ^[2], and ^[3].

In this paper, we consider the following nonlinear swelling soil system with nonlinear source terms of logarithmic-type:

where the constituents $z$ and $u$ represent the displacement of the fluid and the elastic solid material, respectively. The positive constant coefficients $\rho_u$ and $\rho_z$ are the densities of each constituent. The coefficients $a_1, a_2$, and $a_3$ are positive constants satisfying specific conditions. $z_0, z_1, u_0, u_1$ are given data. $\gamma, \beta \geq 0$, $\alpha$ is a small positive constant, and $\nu(.)$ and $\omega(.)$ are the variable functions that are satisfying some specific conditions.

In the present work, our goals are to prove the existence and stability of the system (1.1). We begin by using the Faedo-Galerkin method to prove the local existence of the weak solutions to system (1.1) under suitable assumptions on the variable exponent functions and the logarithmic source terms. We also prove the global existence using the well depth method. Finally, we establish several decay results using the multiplier method and the logarithmic Sobolev inequality.

1.1.   Importance and motivations

Model (1.1) describes swelling of soils with external forces given by nonlinear logarithmic functions. We dissipate this model by the frictional damping mechanism acting on the domain. These dampings of variable exponent-type employ variable exponents in this model, and significantly enhance the ability to capture spatial variations in material properties, nonlinearity, anisotropy, and other complex behaviors. This approach can lead to more accurate simulations and predictions, thereby contributing to the stability, optimization, and design of some tools for a variety of engineering applications ^{[4,5,6,7,8,9]}.

The righthand sides of the system (1.1) represent nonlinear sources of logarithmic-type, which models an external force that amplifies energy and drives the system to possible instability.

We add the logarithmic source terms because they occur in some phenomena; such phenomena are common in nature such as in inflation cosmology, nuclear physics, geophysics, and optics (see ^{[10,11,12,13,14,15,16,17,18,19,20,21,22,23]}).

1.2.   The novelty of our results

In the system (1.1), it is evident that the the damping terms and the source terms are the two major players in this model. Their interactions stimulate many interesting phenomena, which deserve careful investigation. To control an object means to influence its behavior so as to achieve a desired goal. In the system (1.1), the intrinsic frictional damping mechanism acting on the system is responsible for dissipation of its energy. The purpose of this line of study is to find conditions on the initial state to control the dissipations that are needed in order to obtain a decay rate of the energy. In other words, the goal is to discover an adequate choice of the controls that can drive the system from a given initial state to a final given state, in a given time.

The study of the interaction of nonlinear damping and source terms was initiated by Georgiev and Todorova ^[24] in the wave equation. In this line of research, an important breakthrough was made by Bociu and Lasiecka in a series of papers ^[25] and ^[26] where they provided a complete study of a wave equation with damping and supercritical sources in the interior and on the boundary of the domain. Indeed, a source term $\vert u \vert^{m-1} u$ is called subcritical if $1 \leq m < 3$, critical if $m = 3$, and supercritical if $m > 3$, in three space dimensions.

The novelty of our results can be seen from the following aspects:

(1) The source term in our model (1.1) is logarithmic. Let us note here that though the logarithmic nonlinearity is somehow weaker than polynomial nonlinearity, both the existence and stability result are not obtained by straightforward application of the method used for polynomial nonlinearity. We need to make some extra conditions on the nonlinearity coefficient.

(2) The frictional damping mechanismins are nonstandard. They are of variable exponent-type. Variable exponents in the context of swelling soils are often associated with mathematical models used to represent the relationship between soil moisture content and volume change.

(3) How to control the frictional damping mechanism to stabilize the system because the external forces may lead to instability.

1.3.   The originality of the model

The fundamental field equations for the linear theory of swelling porous elastic soils were mathematically presented by Ieşan ^[27] and later simplified by Quintanilla ^[28]. These basic equations are given by

where $z$ and $u$ represent the displacements of the fluid and the elastic solid material, respectively. The coefficients $\rho_z, \rho_u > 0$ and represent the densities of the constituents $z$ and $u$, respectively. The functions $(\phi_1, \chi_1, \psi_1)$ represent the partial tension, internal body forces, and external forces acting on the displacement, respectively. A similar definition holds for $(\phi_2, \chi_2, \psi_2)$, but acts on the elastic solid. Additionally, the constitutive equations of partial tensions are given by

where $a_1, a_3 > 0$ and $a_2 \neq 0$ is a real number. The coefficient matrix $A$ is positive definite, i.e., $a_1 a_3 > a_2^2$. After that, Quintanilla ^[28] investigated

where $\zeta$ is a positive constant, and he obtained an exponential stability result. Similarly, Wang and Guo ^[29] considered

where $\xi(x)$ is an internal viscous damping function with a positive mean. The authors established their exponential stability result by using the spectral method technique. Subsequently, a growing body of new research has explored the stability of system (1.2) by employing various damping mechanisms including viscoelastic damping and/ frictional damping (see, for example ^{[30,31,32,33,34,35,36,37,38,39,40]}). Recently, Al-Mahdi et al. ^[41] established exponential and polynomial decay results for the following system with variable exponent nonlinearity

where the constituents $z$ and $u$ represent the displacement of the fluid and the elastic solid material, respectively. The positive constant coefficients $\rho_u$ and $\rho_z$ are the densities of each constituent. The coefficients $a_1, a_2$, and $a_3$ are positive constants satisfying specific conditions. $z_0, z_1, u_0, u_1$ are given data, and $m(.)$ is a variable function that satisfies some specific conditions.

1.4.   Comparison results

Here, we compare our problem (1.1) with other problems involving source terms of logarithmic-type and source terms of polynomial-type. Regarding swelling soils, many authors investigated the stability analysis of swelling soils problems with different damping mechanism without external forces (source terms). For example, Al-Mahdi et al. ^[42] and ^[43] proved the stability of the swelling soil problem with memory damping terms. Kafini et al. ^[] studied the stability of the swelling soils problem with time delay and variable exponents without source terms.

Logarithmic sources terms have been added in the literature for some other models such as plate equations ^[19], ^[45], and ^[46]. For the polynomial source terms, we refer to the works ^[47], ^[48], and ^[49].

We notice that adding source terms does not improve the stability rate decay. In addition, the logarithmic nonlinearity is weaker than the polynomial nonlinearity. However, we include the logarithmic source terms because they occur in some phenomena. Such phenomena are common in nature such as in inflation cosmology, nuclear physics, geophysics, and optics.

2.   Preliminaries

In this section, we present some preliminaries necessary for proving the stability results. Throughout the paper, $\Omega$ denotes the interval $(0, 1)$ and $c$ represents a generic positive constant.

Let $p:\Omega\rightarrow [1, \infty]$ be a measurable function. The Lebesgue space with a variable exponent $p(\cdot)$ is defined as:

where

Equipped with the following Luxembourg-type norm

the space $L^{p(\cdot)}(\Omega)$ is a Banach space (see ^[50]), separable if $p(\cdot)$ is bounded and reflexive if $1 < p_1 \leq p_2 < \infty,$ where

The variable-exponent Sobolev space is defined as :

This is a Banach space with respect to the norm $\|v\|_{W^{1, p(\cdot)}(\Omega)} = \|v\|_{p(\cdot)}+\|v_x\|_{p(\cdot)}$ and it is separable if $p(\cdot)$ is bounded and reflexive if $1 < p_1 \leq p_2 < \infty$. Furthermore, we set $W^{1, p(.)}_0(\Omega)$ to be the closure of $C^{\infty}_0(\Omega)$ in $W^{1, p(\cdot)}(\Omega)$.

The exponent $p(\cdot): \Omega \rightarrow [1, \infty]$ is said to be satisfying for the log-Hölder continuity condition; that is, if there exists a constant $A > 0$ such that, for all $\delta$ with $0 < \delta < 1,$

Lemma 2.1. ^[50] (Poincaré's inequality) Let $\Omega$ be a bounded domain of $\mathbb{R}^n$ and $p(\cdot)$ satisfies (2.1), then

where the positive constant $c_{\rho}$ depends on $p_1$, $p_2$, and $\Omega$ only. In particular, the space $W_{0}^{1, p(\cdot)}(\Omega)$ has an equivalent norm given by $\|v\|_{W_{0}^{1, p(\cdot)}(\Omega)} = \|v_x\|_{p(\cdot)}.$

Lemma 2.2. ^[50] (Embedding property) Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with a smooth boundary $\partial \Omega.$ Assume that $p, k\in C(\overline{\Omega})$ such that

and $k(x) < p^*(x)$ in $\overline{\Omega}$ with

then we have continuous and compact embedding $W^{1, p(.)}(\Omega)\hookrightarrow L^{k(.)}(\Omega).$ So, there exists $c_e > 0$ such that

For more details about the Lebesgue and Sobolev spaces with variable exponents, ^[50,51,52]. We consider the following hypotheses:

$(A1)$ $\nu, \omega:\overline{\Omega} \rightarrow [1, \infty)$ are measurable functions on $\Omega$ that satisfy the following conditions

where

and they also satisfy the log-Hölder continuity condition; that is, for any $\lambda$ with $0 < \lambda < 1,$ there exists a constant $\delta > 0$ such that,

$(A2)$The coefficients of the system $a_i, \; i = 1, ..., 3$ satisfy $a_1 a_3 -a_2^2 > 0$.

$(A3)$ The constant $\alpha$ in (1.1) satisfies $0 < \alpha < \alpha_0$, where $\alpha_0$ is the positive real number satisfying

where $\tilde{c}$ is a positive constant appearing in (3.7).

Lemma 2.3. ^[14,53] (Logarithmic Sobolev inequality) Let $v$ be any function in $H^{1}_{0}(\Omega)$ and $a > 0$ be any real number, then the following inequality holds:

Remark 2.1. The function $f(s) = \sqrt{\frac{2\pi}{ s}}-e^{-\frac{3}{2}-\frac{1}{s}}$ is continuous and decreasing on $(0, \infty),$ with

Therefore, there exists a unique $\alpha_0 > 0$ such that $f(\alpha_0) = 0$, that is,

Moreover,

Lemma 2.4. ^[54] (Logarithmic Gronwall inequality) Let $c > 0,$ $u\in L^{1}(0, T;\mathbb{R}^+)$, and assume that the function $v:[0, T]\to [1, \infty)$ satisfies

then

The energy functional associated with system (1.1) is defined by

Direct differentiation, using (1.1), gives

Remark 2.2. The nonnegativity of the energy functional is obtained by $(A2)$ and the following identity

Remark 2.3. The following inequality is needed for the proof of our main results:

There exist two positive constants $c_0$ and $d_0$ such that

In fact, $c_0$ is the largest positive constant, which satisfies $c_0\le \frac{\left(A+B\right)^2 }{A^2+B^2}$, and $d_0$ is the smallest positive constant, which satisfies $d_0\ge \frac{\left(A+B\right)^2 }{A^2+B^2}$.

3.   Local existence

First, we multiply the first equation in (1.1) by $\phi \in C^{\infty}_0(\Omega)$ and the second equation by $\psi \in C^{\infty}_0(\Omega)$, integrate each result over $\Omega$, and use Green's formula and the boundary conditions to obtain the definition of the weak solution. Second, we provide a detailed proof of the local existence theorem by using the Faedo-Galerkin approximations.

Definition 3.1. The pair of functions $(z, u)$ is called a weak solution of $\left(P\right),$ if it satisfies the following:

for a.e. $t \in [0, T]$,

$u_t\in L^{\infty}([0, T), L^2(\Omega)) \cap L^\omega(\Omega \times (0, T))$, and the test functions $\phi, \psi \in H^{1}_{0}(\Omega)$. Note that $C^{\infty}_0(\Omega)$ is dense in $H^1_0(\Omega)$. Further, the spaces $H^1_0(\Omega) \subset L^{\nu(.)}(\Omega) \cap L^{\omega(.)}(\Omega)$.

Theorem 3.1. Assume that $(A1)-(A3)$ hold and let $(z_{0}, z_{1}), (u_{0}, u_{1}) \in H^{1}_{0} (\Omega)\times L^{2}(\Omega),$ then problem (1.1) has a unique local weak solution $(z, u)$ on $[ 0, T)$ in the sense of Definition 3.1.

Proof. The proof of the existence of a weak solution of (1.1) consists of four steps:

Step 1. Approximate problem: In this step, we consider $\{w_{j}\}_{j = 1}^{\infty}$ an orthogonal basis of $H_{0}^{1}(\Omega)$ and define, for all $k\geq1,$ ($z^{k}, u^{k})$ a sequence in the finite - dimensional subspace $(V_k \times V_k)$, where $V_{k} = span\{w_{1}, w_{2}, ..., w_{k}\}$ as follows:

for all $x \in \Omega$ and $t \in (0, T)$, satisfying the following approximate problem:

where $\langle \text{ }, \text{ }\rangle$ is the inner product in $L^2(\Omega)$ and

such that

Based on standard existence theory for integro-differential equations, system (3.2) admits a unique local solution $(z^{k}, u^{k})$ on a maximal time interval $[0, T_{k}), \quad 0 < T_{k} < T,$ for each $k\in\mathbb{N}.$

Step 2. A priori estimates: In this step, we show, by priory estimates, that $T_{k} = T$ for each $k\in\mathbb{N}.$ We multiply the first equation by $a_j'(t)$ and the second equation by $b_j'(t)$ in (3.2), sum over $j = 1, 2, ...k$, and add the two equations to obtain

Integration of (3.4) over $(0, t)$ leads to

Using (2.11), Young's inequality, and convergence (3.3), we have

Using (2.12) and applying the logarithmic Sobolev inequality for (3.6), we obtain

where $\tilde{c} = \min{\{c_0 a_1^2, c_0\frac{a_2^2}{a_1}\}},$ $C_0 = cE^k(0)$. Now, we select

and use $(A2)$ to obtain

Combining (3.7) and (3.9), we have

Hence,

Let us note that

Thus, applying the Cauchy-Schwarz' inequality, we get

The addition of the two estimates in (3.12) gives

Combining (3.11) and (3.13) leads to

where, without loss of generality, $C_{1}\ge 1$. The logarithmic Gronwall inequality implies that

and hence,

After combining (3.10) and (3.15), we obtain

Therefore, the local solution $(z^{k}, u^{k})$ of system (3.2) can be extended to $(0, T)$, for all $k\geq 1.$ Furthermore, we have

Consequently, we have, up to two subsequences,

Step 3. The logarithmic terms: In this step, we show that the approximate solutions $(z^k, u^k)$ satisfy for all $k\ge 1$,

Making use of the arguments in (3.16) and applying the Aubin-Lions theorem, we find, up to subsequences, that

and

Using (3.18), and the fact that the map $s \to s\ln{\vert s\vert}^\alpha$ is continuous on $\mathbb{R}$, then we have the convergence

Using the embedding of $H^1_0(\Omega)$ in $L^{\infty}(\Omega)$ (since $\Omega\subset \mathbb{R}$), it is clear that $z^k \ln{{\vert z^k\vert}^\alpha}$ is bounded in $L^{\infty}(\Omega\times(0, T))$. Next, taking into account the Lebesgue bounded convergence theorem ($\Omega$ is bounded), we get

Similarly, we can establish the second argument of (3.17).

Step 4. The nonlinear terms: In this step, we show that

and that $(z, u)$ satisfies the partial differential equations of (1.1) on $\Omega \times (0, T).$

Since $(z_{t}^{k})$ is bounded in $L^{\nu(.)}(\Omega \times (0, T)),$ then $({\vert z_{t}^{k}\vert}^{\nu(.)-2}z_{t}^{k})$ is bounded in $L^{\frac{\nu(.)}{\nu(.)-1}}(\Omega \times (0, T)).$ Hence, up to a subsequence,

Similarly, we have

We can show that

by following the same steps as in ^[55,56]. Now, integrate (3.2) on $(0, t)$ to obtain $\forall j < k$,

Using all the above convergence and taking $k \to+\infty$, we easily check that $\forall j < k$,

Consequently, we have $\forall w\in H^1_0(\Omega)$

All terms define absolute continuous functions, so we get, for a.e. $t\in [0, T]$ and $\forall w\in H^1_0(\Omega)$,

This implies that

This implies that $(z, u)$ satisfies the two differential equations in (1.1), on $\Omega \times (0, T)$.

Step 5. The initial conditions: We can handle the initial conditions like the one in ^[55]. Hence, we deduce that $(z, u)$ is the unique local solution of (1.1). This completes the proof of Theorem 3.1. □

4.   Global existence

By using the potential wells, we prove the existence of the global solution to our problem. To this end, we define the following functionals:

Remark 4.1. (1) From the above definitions, it is clear that

(2) According to the logarithmic Sobolev inequality, $J(z, u)$ and $I(z, u)$ are well-defined.

We define the potential well (stable set):

The potential well depth is defined by

and the well-known Nehari manifold is

Proceeding as in ^[57,58], one has

Lemma 4.1. For any $(z, u)\in H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega)$, $\| z\|_{2}\ne 0$, and $\| u\|_{2}\ne 0$, let $g(\lambda) = J(\lambda z, \lambda u)$, then we have

where

where $\alpha_0 = \left(a_3-\frac{a_2^2}{a_1}\right) > 0$.

Proof.

Since $\| z\|_{2}\ne 0$ and $\| u\|_{2}\ne 0$, then $g(0) = 0,$ $g(+\infty) = -\infty$, and

which implies that $\frac{d}{d\lambda}J(\lambda z, \lambda u)_{\lambda = \lambda^*} = 0,$ $J(\lambda z, \lambda u)$ is increasing on $0 < \lambda\le \lambda^*$, decreasing on $\lambda^*\le \lambda < \infty$, and reaching its maximum value at $\lambda = \lambda^*$. In other words, there exists a unique $\lambda^* \in(0, \infty)$ such that $I(\lambda^* z, \lambda^* u) = 0$, which establishes the desired result.□

Lemma 4.2. Let $(z, u)\in H^1_0(\Omega)\times H^1_0(\Omega)$ and $\beta_0 = \sqrt{\frac{2\pi \tilde{c}}{\alpha}}e^{1+\frac{1}{\alpha}}$. If $0 < \|z\|_2\le \beta_0$ and $0 < \|u\|_2\le \beta_0$, then $I(z, u)\ge 0$.

Proof. Using the logarithmic Sobolev inequality (2.4), for any $a > 0$, we have

Taking $a < \min \left\{\sqrt{\frac{2\pi \tilde{c}}{\alpha}}, \sqrt{\frac{2\pi \tilde{c}}{\alpha}} \right\}$ in (4.8), we obtain

If $0 < \|z\|_2\le \beta_0$ and $0 < \|u\|_2\le \beta_0$, then

which gives $I(z, u)\ge 0$. □

Lemma 4.3. The potential well depth $d$ satisfies

Proof. The proof of this lemma is similar to the proof of Lemma 4.3. in ^[59].

□

Lemma 4.4. Let $(z_{0}, z_{1}), (u_{0}, u_{1})\in H_{0}^{1}(\Omega)\times L^2(\Omega)$ such that $0 < E(0) < d$ and $I(z_0, u_0) > 0$, then any solution of (1.1) is $(z, u)\in W$.

Proof. Let $T$ be the maximal existence time of a weak solution of $(\psi, \varphi)$. From (2.10) and (4.4), we have

then we claim that $(z(t), u(t))\in W$ for all $t\in [0, T)$. If not, then there is a $t_0 \in (0, T)$ such that $I(z(t_0), u(t_0)) < 0$. Using the continuity of $I(z(t), u(t))$ in $t$, we deduce that there exists a $t_*\in (0, T)$ such that $I(z(t_*), u(t_*)) = 0$. Using the definition of $d$ in (4.5) gives

which is a contradiction. □

5.   Stability

In this section, we state and prove our main decay results. For this purpose, we present the following lemmas.

Lemma 5.1. For any $\eta > 0$, we have the following:

and if $1 < \omega_1 < 2$, we have

Lemma 5.2. For any $\lambda > 0$, we have the following:

and if $1 < \nu_1 < 2$, we have

Proof. We prove Lemma 5.1, and the proof of Lemma 5.2 will be similar. We start by applying Young's inequality with $\xi(x) = \frac{\omega(x)}{\omega(x)-1}$ and $\xi^{\prime}(x) = \omega(x)$. So, for a.e $x\in (0, 1)$ and any $\eta > 0$, we have

where

Hence,

Next, using (2.9), (2.10), (4.8), Poincaré's inequality, and the embedding property, we get

where $c_e$ is the embedding constant,

and

Thus, from (5.5) and (5.6), we find that

Combining all the above estimations, estimate (5.1) is established. To prove (5.2), we set

then, we have

We notice that on $\Omega_1$, we have

Therefore, by using Young's and Poincaré's inequalities and (5.10), we find that

where

Next, by the case of $\omega(x) \ge 2,$ we have

Combining (5.11) and (5.13), the proof of (5.2) is completed. □

Remark 5.1. For the stability results, we assume that the coefficients $a_i, \; i = 1, ..., 3$ satisfy

It is clear that (5.14) gives the condition in $(A2)$.

Lemma 5.3. Assume that $(A1-A3)$ and (5.14) hold and let $(z_0, z_1), (u_0, u_1) \in H^{1}_{0}(\Omega) \times L^{2}(\Omega)$. Assume further that $0 < E(0) < \ell \tau < d$, where

then the functional

satisfies, along with the solutions of (1.1) and for a suitable choice of $N$,

and

where $\alpha_1 = \frac{2-\omega_1}{\omega_1-1} > 0, \alpha_2 = \frac{2-\nu_1}{\nu_1-1} > 0, \alpha_3 = \frac{2-m_1}{m_1-1} > 0$, and $m_1 = \min\{\nu_1, \omega_1\}$.

Proof. If we want to prove all cases, the proof will be very lengthy, so we prove (5.17)$_2$ and the proofs of the other cases are very similar with minor modifications. To prove (5.17)$_2$, we differentiate $L(t)$ and use integrations by parts, to get

Using Young's inequality, we have for some positive constants $\lambda_i$,

and

Using (5.2), (5.4), and (5.18)–(5.21), we have

Using (2.10) and the logarithmic Sobolev inequality, (5.22) becomes

Now, we select $N$ large enough so that $N-c > 0,$ then we select $a < \sqrt{\frac{2\pi a_0}{\alpha}}$, where $a_0 = \min{\{a_1, a_3\}}$, which makes

After that, we choose $\eta = \frac{a_3-\frac{ \alpha a^2}{2\pi}}{2c\beta}$, and $\frac{2a_2^2}{a_1-\frac{ \alpha a^2}{2\pi}} < \lambda_1 < \frac{a_3-\frac{ \alpha a^2}{2\pi}}{2}$, to get

This selection is possible thanks to (5.14). Using (2.9), (2.10), and the fact that $u\in W$,

After taking $a$ satisfying

and $\lambda_4$ is small enough, we guarantee the following:

Then, (5.23) reduces to

Using Young's inequality with $\zeta = \frac{1}{\omega_1-1}$ and $\zeta^* = \frac{1}{2-\omega_1}$, for any $\varepsilon > 0$, we estimate this term $E^\alpha(t)\big(-E'(t)\big)^{\omega_1-1}$ as follows:

Multiplying both sides of the last inequality by $E^{-\alpha}$, where $\alpha = \frac{2-\omega_1}{\omega_1-1}$, gives us

Inserting this estimate in the last term in (5.25), we find that

By taking $\varepsilon$ small enough and using the nonincreasing property of $E$, (5.17) is established. On the other hand, we can choose $N$ even larger (if needed) so that $L \sim E$. □

Lemma 5.4. Assume that $(A1)$ holds, then

and

Proof. By recalling (2.10), it is easy to establish (5.27). To prove the first estimate in (5.28), we set the following partitions

The use of Hölder's and Young's inequalities and (2.9), give for $\Omega _{1}$,

and for $\Omega _{2}$, we get

Combining (5.30) and (5.31), the first estimate in (5.28) is established, and we repeat the same steps to establish the second estimate in (5.28). □

Theorem 5.1. Assume that $(A1-A3)$ hold and $\nu_1, \omega_1 \geq 2$, then the energy functional (2.9) satisfies, for some positive constants $\lambda_i, \sigma_i, \mu_i > 0, \; i = 1, 2, 3$, and for any $t\geq 0$,

and

where $m_2 = \min\{\nu_2, \omega_2\}$.

Proof. To prove $(5.32)_1$, we impose Lemma (5.4) in $(5.17)_1$ to obtain

which leads to

where $L_1 = L+c E \sim E$. Integrating (5.35) over $\left(0, t\right)$ and using the fact that $L_1, L \sim E$, the proof of $(5.32)_1$ is finished, and the remaining proofs of $(5.32)_2$ and $(5.32)_3$ can be achieved in the same way. Now, it is enough to prove the estimate given in $(5.33)_3$, and the remaining can be achieved in the same way. To this end, we also apply Lemma 5.4 in Eq $(5.17)_1$ to have

By multiplying (5.36) by $E^{\alpha}$, where $\alpha = \frac{\nu_2-2}{2} > 0$, we get

Applying Young's inequality twice in (5.37), we find that for $\varepsilon > 0$,

We discuss two cases:

Case 1. If $\nu_2 < \omega_2$, we will have

Using the fact that $E' \leq 0$, we get

Choosing $\varepsilon$ small enough, we see that (5.39) becomes

where $L_2 = E^{\alpha} L+cE \sim E$. By integrating (5.40) over $(0, t)$ and using the fact that $E \sim L_2$, we obtain

where $\alpha = \frac{\nu_2-2}{2}$.

Case 2. If $\omega_2 < \nu_2$, in this case we get

where $\alpha = \frac{\omega_2-2}{2}$. So, the proof of $(5.33)_3$ can be completed by taking $m_2 = \min\{\nu_2, \omega_2\}$. □

Theorem 5.2. Assume that $(A1-A3)$ hold, $1 < \nu_1, \omega_1 < 2$ and, $\nu_2 = \omega_2 = 2$, then the energy functional (2.9) satisfies, for a positive constants $C_i, \; i = 1, 2, 3$, and for any $t > 0$,

where $m_1 = \min\{\nu_1, \omega_1\}$.

Proof. To prove $(5.43)_1$, we impose Lemma (5.4) in $(5.17)_2$ to get

where $\alpha_1 = \frac{2-\omega_1}{\omega_1-1} > 0$. By taking $L_1 = L+cE \sim E$, this becomes

Multiplying (5.44) by $E^{\alpha_1}$, we have

By taking $L_2 = E^{\alpha}L_1+cE \sim E$, this becomes

Therefore, we obtain the following decay estimate

where $\alpha_1 = \frac{2-\omega_1}{\omega_1-1}$. The proof of $(5.43)_1$ is completed, and the proof of $(5.43)_2$ and $(5.43)_3$ can be achieved in the same way.□

Theorem 5.3. Assume that $(A1-A3)$ hold, $1 < \nu_1, \omega_1 < 2$, and $\nu_2, \omega_2 > 2$, then the energy functional (2.9) satisfies, for a positive constants $C_i, \; i = 1, 2, 3$, and for any $t > 0$,

where $m_2 = \min\{\nu_2, \omega_2\}$

Proof. To prove $(5.46)_1$, we impose Lemma (5.4) in $(5.17)_1$ to get

where $\alpha_1 = \frac{2-\omega_1}{\omega_1-1} > 0$. Multiplying by $E^{\alpha}$ where $\alpha = \frac{\omega_2-2}{2} > 0$, and using $\alpha-\alpha_1 > 0$ and Young's inequality twice, we obtain, for $\varepsilon > 0$,

Assuming that $\omega_2 > \nu_2$,

Using the fact that $E$ is nonincreasing, we obtain

Taking $\varepsilon$ small enough, the above estimate becomes:

where $L_2 = E^{\alpha}L+cE \sim E$. Integration (5.47) over $(0, t)$ and using $E \sim L_2$, we get

where $\alpha = \frac{\omega_2-2}{2}$. So, the proof of $(5.46)_1$ is completed and the proofs of $(5.46)_2$ and $(5.46)_3$ will be the same. □

6.   Conclusions and open problems

In this paper, we proved the local existence result of solutions of the nonlinear swelling porous-elastic system by using the Faedo-Galerkin method. Furthermore, we proved the global existence of solutions by using the well-depth method. Finally, we established several decay results by employing the multiplier method and the logarithmic Sobolev inequality. The problem will be very interesting if we consider the damping condiments $\gamma$ and $\beta$ as functions of $x$ and $t$, i.e, $\gamma = \gamma(x, t)$ and $\beta = \beta(x, t)$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to acknowledge the support provided by King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia. The support provided by the Interdisciplinary Research Center for Construction & Building Materials (IRC-CBM) at King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia, for funding this work through Project No. INCB2402, is also greatly acknowledged.

Conflict of interest

The authors declare no competing interests.