EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6073 ISSN 1307-5543 – ejpam.com Published by New York Business Global Controllability of a System with Nonlinear Damping Devices and Nonlinear Source Terms in Elasticity Problems: Existence, Time Blow-up, and Numerical Results Adel M. Al-Mahdi1,2,∗, MohammedM. Al-Gharabli1,2, Mohammed D. Kassim3, Abdelaziz Soufyane4, Mostafa Zahri4, Soh Edwin Mukiawa5 1 Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 2 The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 3 Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 34151, Dammam, Saudi Arabia 4 Department of Mathematics and Statistics, University of Sharjah, Sharjah, United Arab Emirates 5 Department of Mathematics,University of Hafr Al Batin, Hafar Al Batin 39524, Saudi Arabia Abstract. Swelling soil problems arise in various real-world applications, such as geomechanics, biomedical engineering, and hydrogel-based materials, where fluid interaction with elastic struc- tures influences mechanical stability. In this study, we investigate a swelling soil system incorpo- rating two nonlinear variable exponent damping and source terms, which provide a more adaptable framework for capturing heterogeneous material behaviors and evolving energy dissipation mech- anisms. Using the Faedo-Galerkin method and the Banach Contraction Theorem, we establish the local existence and uniqueness of weak solutions under suitable conditions on the variable exponent functions. Furthermore, we demonstrate the global existence of solutions and identify conditions leading to finite-time blow-up, offering insights into stability and failure prediction in porous-elastic media. To validate our theoretical findings, we present numerical simulations illus- trating the blow-up behavior, emphasizing the role of variable exponent damping in influencing system dynamics. 2020 Mathematics Subject Classifications: 93D20, 35B40 Key Words and Phrases: Swelling soil problems, Faedo-Galerkin method, Banach Contraction Theorem, Blow-up, Variable exponents, Numerical methods ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6073 Email addresses: almahdi@kfupm.edu.sa (A. M. Al-Mahdi), mahfouz@kfupm.edu.sa (M. M. Al-Gharabli), mdkassim@iau.edu.sa (M. D. Kassim), asoufyane@sharjah.ac.ae (A. Soufyane), mzahri@sharjah.ac.ae (M. Zahri), mukiawa@uhb.edu.sa (S. E. Mukiawa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 2 of 29 1. Introduction Swelling soils, also known as expansive soils, are characterized by an increase in volume when exposed to moisture. The clay minerals within these soils naturally attract and ab- sorb water. As described succinctly by Handy [1], ”when water is introduced to swelling soils, the water molecules are pulled into gaps between the soil plates. As more water is absorbed, the plates are forced further apart, leading to an increase in soil pore pres- sure.” Consequently, swelling soils pose significant geotechnical and structural challenges, impacting both the environment and society, as illustrated in Figures (1)-(2). Figure 1: Cracking in soil Figure 2: Cracking in a building structure Swelling soils are found worldwide. As reported by Nelson and Miller [2], the American Society of Civil Engineers estimates that one in four homes suffer damage caused by expansive soils. Typically, such soils result in greater financial losses for property owners than earthquakes, floods, hurricanes, and tornadoes combined. Therefore, it is crucial to explore effective methods to mitigate or eliminate the damage caused by swelling soils. Further theoretical background and details can be found in [3–5]. Although various chemical admixtures and other costly methods have been employed to mitigate swelling soil damage, these solutions are often ineffective in the long term. This research aims to stabilize swelling soils by utilizing damping mechanisms that are both effective and environmentally sustainable. Mathematically, swelling soil models consist of two coupled partial differential equations that describe the displacement of both the fluid and the elastic solid material. To the best of our knowledge, the swelling soil system was first proposed by Ieş [6] and later simplified by Quintanilla [7]. The fundamental field equations for the linear theory of swelling soils are given by { ρzztt = P1x −G1 + F1, ρuutt = P2x +G2 + F2, (1) where the variables z and u represent the displacement of the fluid and the elastic solid material, respectively. The positive constants ρz and ρu denote the densities of each constituent. The terms (P1, G1, F1) correspond to the partial tension, internal body forces, A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 3 of 29 and external forces acting on the displacement, respectively. Similarly, (P2, G2, F2) have analogous definitions for the elastic solid. The constitutive equations of partial tensions are expressed as[ P1 P2 ] = [ a1 a2 a2 a3 ] ︸ ︷︷ ︸ A [ zx ux ] , (2) where a1, a3 are positive constants and a2 ̸= 0 is a real number. The matrix A is positive definite in the sense that a1a3 > a22. Quintanilla [7] investigated the system:{ ρzztt = a1zxx + a2uxx − ξ(zt − ut) + a3zxxt, ρuutt = a2zxx + a3uxx + ξ(zt − ut), (3) where ξ is a positive coefficient. Under initial and homogeneous Dirichlet boundary con- ditions, he established an exponential stability result. Similarly, Wang and Guo [8] con- sidered: { ρzztt = a1zxx + a2uxx − ρzγ(x)zt, ρuutt = a2zxx + a3uxx, (4) where γ(x) represents an internal viscous damping function with a positive mean. Using spectral analysis, they established an exponential stability result. Several recent studies have introduced new stabilization mechanisms for swelling soil models (1) that are effective, economical, and environmentally sustainable [9–12, 12]. In recent years, there has been growing interest in treating equations with variable exponent nonlinearities due to their applications in the mathematical modeling of non-Newtonian fluids. One prominent example is electro-rheological fluids, which can undergo drastic changes under the influence of external electromagnetic fields. In these cases, the vari- able exponent nonlinearity depends on physical parameters such as density, temperature, saturation, and electric field. For more results in this direction, we refer to [13–21]. Recently, Al-Mahdi et al. [22] established exponential and polynomial decay results for swelling soils governed by the system:{ ρzztt − a1zxx − a2uxx + α|zt|m(·)−2zt = β|z|m(·)−2z, in (0, 1)× (0,∞), ρuutt − a3uxx − a2zxx = 0, in (0, 1)× (0,∞), (5) under suitable conditions on the variable exponents. However, an important question arises: Does system (5) with γ, β > 0 admit global existence, stability, or blow-up results under certain conditions on the variable exponents? In this paper, we address this question by considering the following swelling soil system of the form: A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 4 of 29  ρzztt − a1zxx − a2uxx + γ|zt|p(·)−2zt = c|z|m(·)−2z, in Ω× (0,∞), ρuutt − a3uxx − a2zxx + β|ut|q(·)−2ut = d|u|ℓ(·)−2u, in Ω× (0,∞), u(x, 0) = u0(x), ut(x, 0) = u1(x), z(x, 0) = z0(x), zt(x, 0) = z1(x) x ∈ Ω, z(0, t) = z(1, t) = u(0, t) = u(1, t) = 0 t ≥ 0, (6) where Ω denotes the interval (0, 1). The positive constant coefficients ρu and ρz are the densities of each constituent. The coefficients a1, a2 and a3 are positive constants satis- fying specific conditions. The coefficients γ, β, c, d > 0, z0, z1, u0, u1 are given data and p(.), q(.),m(.), ℓ(.) are function satisfying some conditions to be specified in the next sec- tion. System (6) exhibits several key differences from previously studied swelling porous-elastic models. Unlike classical models, where damping and source terms are typically governed by constant power-law exponents, our system introduces variable exponent functions p(x), q(x), m(x), and ℓ(x). This formulation provides a more flexible and generalized framework for capturing energy dissipation and nonlinear interactions, accommodating spatial het- erogeneity in material properties. As a result, our model is more adaptable to real-world applications, including soil swelling, biomechanics, and geomechanics. The presence of variable exponent damping and source terms introduces new challenges in the global existence and the blow-up analysis. We establish conditions under which solutions exhibit finite-time blow-up, extending classical results from constant exponent cases to more general variable exponent settings. These advancements mark a significant generalization of existing studies on swelling porous-elastic systems. By incorporating variable exponent nonlinearities, our work pro- vides novel insights into the behavior of complex porous-elastic media under nonlinear stress conditions. First, we establish the existence and uniqueness results of a weak solution and then we prove the global existence of the solutions under suitable assumptions on the variable exponents. Second, we show that the solutions with negative-initial energy blow-up in a finite time. Finally, we produce some numerical tests and examples to illustrate our blow-up results. The interaction between the damping and the source terms was first considered by Levine [23, 24] in the linear damping case, (m = 2), in the following wave equation utt −∆u+ a|ut|m−2ut = b|u|p−2u,in Ω, t > 0, (7) where a, b > 0, p > 2, m ≥ 1, Ω is a bounded domain in Rn. He showed that solutions with negative initial energy blow up in finite time. Then, Georgiev and Todorova [25] extended Levine’s result to the nonlinear damping case (m > 2). In their work, the A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 5 of 29 authors introduced a different method and determined suitable relations between m and p for which there is global existence or alternatively finite time blow up. More precisely: they showed that solutions with any initial data continue to exist globally (in time) if m ≥ p and blow up in finite time if m < p and the initial energy is sufficiently negative. Without imposing the condition that the initial energy is sufficiently negative, Messaoudi [26] extended the blow up result of [25] to solutions with negative initial energy only. Messaoudi [27]considered the following nonlinear viscoelastic wave equation utt −∆u+ ∫ t 0 g(t− s)∆u(s)ds+ a|ut|m−2ut = b|u|p−2u, x ∈ Ω, t > 0, (8) where Ω is a bounded domain of Rn. He proved that any weak solution with negative initial energy blows-up in finite time if p > m. Also the case of a stronger damping is considered and it is showed that solutions exist globally for any initial data, in the appropriate space, provided that m ≤ p. For more results in blow-up, we refer the reader to see [28–32] and the references therein. 2. Preliminaries In this section, we present some material needed in the proof of our results. Throughout this paper Ω = (0, 1) and C is used to denote a generic positive constant. • (A1): p, q,m, ℓ : Ω → [1,∞) are measurable functions on Ω satisfying all the follow- ing conditions 2 ≤ p1 ≤ p(x) ≤ p2 < m1 ≤ m(x) ≤ m2 <∞, 2 ≤ q1 ≤ q(x) ≤ q2 < ℓ1 ≤ ℓ(x) ≤ ℓ2 <∞, where p1 := essinfx∈Ωp(x), p2 := esssupx∈Ωp(x), m1 := essinfx∈Ωm(x), m2 := esssupx∈Ωm(x), q1 := essinfx∈Ωq(x), q2 := esssupx∈Ωq(x), ℓ1 := essinfx∈Ωℓ(x), ℓ2 := esssupx∈Ωℓ(x). and satisfy the log-Hölder continuity condition; that is for any δ with 0 < δ < 1, there exists a constant A > 0 such that, |f(x)− f(y)| ≤ − A log |x− y| , for all x, y ∈ Ω, with |x− y| < δ. (9) • (A2): The coefficients ai, i = 1, ..., 3 satisfy a1a3 − a22 > 0. 3. Technical Lemmas In this section, we present and establish some lemmas needed for the proof of our main results. A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 6 of 29 Lemma 1. The energy of the problem (6) is defined by E(t) = 1 2 ∫ Ω [ ρzz 2 t + ρuu 2 t + a3u 2 x + a1z 2 x + 2a2zxux ] dx −c ∫ Ω |z|m(x) m(x) dx− d ∫ Ω |u|ℓ(x) ℓ(x) dx, (10) and satisfies the following E′(t) = −γ ∫ Ω |zt|p(·)dx− β ∫ Ω |ut|q(·)dx ≤ 0. (11) Proof. Multiplying the equations in (6) by zt and ut respectively and then integrating over the interval Ω = (0, 1) to get ρz ∫ Ω ztzttdx− a1 ∫ Ω ztzxxdx− a2 ∫ Ω ztuxxdx+ γ ∫ Ω zt|zt|p(·)−2ztdx + ρu ∫ Ω ututtdx− a3 ∫ Ω utuxxdx− a2 ∫ Ω utzxxdx+ β ∫ Ω ut|ut|q(·)−2utdx = c ∫ Ω zt|z|m(·)−2zdx+ d ∫ Ω ut|u|ℓ(·)−2udx. (12) Using the integration by parts and the boundary conditions and summing up all the results, Eq. (12) becomes ρz ∫ Ω ztzttdx+ a1 ∫ Ω zxzxtdx+ a2 ∫ Ω zxtuxdx+ γ ∫ Ω zt|zt|p(·)−2ztdx ρu ∫ Ω ututtdx+ a3 ∫ Ω uxtuxdx+ a2 ∫ Ω uxtzxdx+ β ∫ Ω ut|ut|q(·)−2utdx = c ∫ Ω zt|z|m(·)−2zdx+ d ∫ Ω ut|u|ℓ(·)−2udx. (13) Using the following differential equations: ρz ∫ Ω ztzttdx = ρz 2 d dt ∫ Ω z2t dx, ρu ∫ Ω ututtdx = ρu 2 d dt ∫ Ω u2tdx, a1 ∫ Ω zxzxtdx = a1 2 d dt ∫ Ω z2xdx, a3 ∫ Ω uxuxtdx = a3 2 d dt ∫ Ω u2xdx, a2 ∫ Ω (zxtux + uxtzx) dx = a2 d dt ∫ Ω uxzxdx, c ∫ Ω zt|z|m(·)−2zdx = c d dt ∫ Ω |z|m(x) m(x) dx, and d ∫ Ω ut|u|ℓ(·)−2udx = d d dt ∫ Ω |u|ℓ(x) ℓ(x) dx. (14) Combing (14) and (13), we obtain A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 7 of 29 1 2 d dt ∫ Ω [ ρzz 2 t + ρuu 2 t + a3u 2 x + a1z 2 x + 2a2zxux ] dx− d dt [ c ∫ Ω |z|m(x) m(x) dx− d ∫ Ω |u|ℓ(x) ℓ(x) dx ] = −γ ∫ Ω |zt|p(·)dx− β ∫ Ω |ut|q(·)dx. This gives d dt E(t) = E′(t) = −γ ∫ Ω |zt|p(·)dx− β ∫ Ω |ut|q(·)dx ≤ 0, where E(t) is defined in (10) and this completes the proof of (11). Lemma 2. For any z, u ∈ H1 0 (Ω) and p(·), q(·) satisfying (A1), we have∫ Ω |z|p(x)dx ≤ Cp1 e ||zx||p12 + Cp2 e ||zx||p22∫ Ω |u|q(x)dx ≤ Cq1 e ||ux||q12 + Cq2 e ||ux||q22 , (15) where Ce is the embedding constant. Proof. The proof of this lemma can be found in [33]. As in [33], we have the following: ϱ(z) := ∫ Ω |z|m(x)dx ≥ C||z||m1 m1 , (16) and ϱ(u) := ∫ Ω |u|ℓ(x)dx ≥ C||u||p1p1 . (17) Lemma 3. Assume that (A1) holds. Then, we have∫ Ω |z|p(x)dx ≤ C ( ϱ(z) p1 m1 + ϱ(z) p2 m1 ) ,∫ Ω |u|q(x)dx ≤ C ( ϱ(u) q1 ℓ1 + ϱ(u) q2 ℓ1 ) . (18) Proof. The proof of this lemma can be found in [33]. A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 8 of 29 4. Local Existence In this section, we give a detailed proof of the local existence theorem by using the Faedo- Galerkin approximations and the Banach-Fixed-point theorem. We multiply the first equation in (6) by ϕ ∈ C∞ 0 (Ω) and the second equation by ψ ∈ C∞ 0 (Ω), integrate each result over Ω, use Green’s formula and the boundary conditions to obtain the following definition: Definition 1. Let T > 0. Any pair of functions z, u ∈ L∞([0, T ), H1 0 (Ω)), zt ∈ L∞([0, T ), L2(Ω)) ∩ Lp(Ω× (0, T )) and ut ∈ L∞([0, T ), L2(Ω)) ∩ Lq(Ω× (0, T )) is called a weak solution of system (6), if d dt ∫ Ω ρzztϕ(x)dx+ a1 ∫ Ω zxϕx(x)dx+ a2 ∫ Ω uxϕx(x)dx +γ ∫ Ω |zt|p(.)−2ztϕ(x)dx = c ∫ Ω |z|m(.)−2zϕ(x)dx d dt ∫ Ω ρuutψ(x)dx+ a3 ∫ Ω uxψx(x)dx+ a2 ∫ Ω zxψx(x)dx +β ∫ Ω |ut|q(.)−2utψ(x)dx = d ∫ Ω |u|ℓ(.)−2uψ(x)dx z(0) = z0, zt(0) = z1, u(0) = u0, ut(0) = u1, (19) for a.e. t ∈ [0, T ] and all test functions ϕ, ψ ∈ H1 0 (Ω). Note that C∞ 0 (Ω) is dense in H1 0 (Ω). In addition, the spaces H1 0 (Ω) ⊂ Lp(.)(Ω) ∩ Lq(.)(Ω), under the conditions (A1) and (A2). Before establishing the existence theorem of a local weak solution of problem (6), we first consider, the following initial-boundary-value problem: ρzztt − a1zxx − a2uxx + γ|zt|p(·)−2zt = f(x, t), in Ω× (0,∞), ρuutt − a3uxx − a2zxx + β|ut|q(·)−2ut = g(x, t), in Ω× (0,∞), z(0, t) = z(1, t) = u(0, t) = u(1, t) = 0 t ≥ 0 , (z(0), u(0)) = (z0, u0), (zt(0), ut(0)) = (z1, u1), in Ω, (Q) where f, g ∈ L2 (Ω× (0, T )) and (u0, u1), (v0, v1) ∈ H1 0 (Ω)× L2(Ω). Theorem 1. Assume that (A1) and (A2) hold and let (u0, u1), (v0, v1) ∈ H1 0 (Ω)×L2(Ω), then problem (Q) has a unique local weak solution (u, v) on [0, T ). Proof. UNIQUENESS: Suppose that (Q) has two solutions (z1, u1) and (z2, u2). Then, (z, u) = (z1−z2, u1−u2) satisfies, in the sense of distribution, the following problem: ρzztt − a1zxx − a2uxx + γ |z1t|p(x)−2 z1t − γ |z2t|p(x)−2 z2t = 0, in Ω× (0,∞), ρuutt − a3uxx − a2zxx + β |u1t|q(x)−2 u1t − β |u2t|q(x)−2 u2t = 0, in Ω× (0,∞), z(0, t) = z(1, t) = u(0, t) = u(1, t) = 0 t ≥ 0 , (z(0), u(0)) = (z0, u0), (zt(0), ut(0)) = (z1, u1), in Ω. A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 9 of 29 Multiplying the first differential equation by zt and the second by ut and then integrating the result over Ω, we obtain d dt [ ρz 2 ||zt||22 + ρu 2 ||ut||22 + a1 2 ||zx||22 + a3 2 ||ux||22 + a2 ∫ Ω uxzxdx ] + γ ∫ Ω ( |z1t|p(x)−2 z1t − |z2t|p(x)−2 z2t ) (z1t − z2t)dx + β ∫ Ω ( |u1t|q(x)−2 u1t − |u2t|q(x)−2 u2t ) (u1t − u2t)dx = 0. (20) Integrating (20) over (0, t), to get ρz 2 ||zt||22 + ρu 2 ||ut||22 + a1 2 ||zx||22 + a3 2 ||ux||22 + a2 ∫ Ω uxzxdx + γ ∫ t 0 ∫ Ω ( |z1t|p(x)−2 z1t − |z2t|p(x)−2 z2t ) (z1t − z2t)dxds + β ∫ t 0 ∫ Ω ( |u1t|q(x)−2 u1t − |u2t|q(x)−2 u2t ) (u1t − u2t)dxds = 0. (21) By using the following inequality[ |Y |b(x)−2Y − |Z|b(x)−2Z ] (Y − Z) ≥ 0, b(x) ≥ 2, (22) for all x ∈ Ω and Y,Z ∈ R, we have ρz 2 ||zt||22 + ρu 2 ||ut||22 + a1 2 ||zx||22 + a3 2 ||ux||22 + a2 ∫ Ω uxzxdx ≤ 0. (23) Applying the following Cauchy-Schwarz’ inequality: | ⟨v1, v2⟩ | ≤ ∥v1∥∥v2∥, ∀v1, v2 ∈ L2(0, 1), (24) and the following Young inequality: |ab| ≤ 1 2 ( ϵa2 + 1 ϵ b2 ) , ∀a, b ∈ R, ∀ϵ > 0, (25) we see that, for any ϵ > 0, a3 ∥ux∥2 + a1 ∥zx∥2 + 2a2 ∫ Ω uxzxdx ≥ ( a3 − |a2| ϵ ) ∥ux∥2 + (a1 − |a2|ϵ) ∥zx∥2 , by choosing ϵ = 1 2|a0| ( a2 − a1 + √ (a1 − a3)2 + 4a22 ) (ϵ is well defined and positive, since a2 ̸= 0), we obtain a3 ∥ux∥2 + a1 ∥zx∥2 + 2a2 ∫ Ω uxzxdx ≥ C̃ ( ∥ux∥2 + ∥zx∥2 ) , (26) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 10 of 29 where C̃ := 1 2 ( a1 + a3 − √ (a1 − a3)2 + 4a22 ) . Combining (23) and (26), we find ρz ∥zt∥22 + a1 ∥zx∥22 = 0. Similarly, we obtain ρu ∥ut∥22 + a3 ∥ux∥22 = 0. Therefore, zt(x, .) = ut(x, .) = 0 on Ω and ux(., t) = zx(., t) = 0, for a.e t ∈ (0, T ). This implies u = z = 0 on Ω×(0, T ), since u = z = 0 on ∂Ω×(0, T ). This proves the uniqueness. EXISTENCE: The proof of the existence of a weak solution of (Q) consists of four steps: Step 1. Approximate problem: In this step, we consider {wj}∞j=1 an orthogonal basis of H1 0 (Ω) and define, for all k ≥ 1, (zk, uk) a sequence in the finite - dimensional subspace (Vk × Vk), where Vk = span{w1, w2, ..., wk}, as follows: zk(x, t) = k∑ j=1 aj(t)wj , uk(x, t) = k∑ j=1 bj(t)wj , for all x ∈ Ω and t ∈ (0, T ) satisfying the following approximate problem: ρz⟨zktt, wj⟩L2(Ω) + a1⟨zxk, wjx⟩L2(Ω) + a2⟨ukx, wjx⟩L2(Ω) + γ⟨|zkt | p(x)−2 zkt , wj⟩L2(Ω) = ⟨f(x, t), wj⟩L2(Ω), j = 1, 2, ..., k, ρu⟨uktt, wj⟩L2(Ω) + a3⟨uxk, wjx⟩L2(Ω) + a2⟨zkx, wjx⟩L2(Ω) + β⟨|ukt | q(x)−2 ukt , wj⟩L2(Ω) = ⟨g(x, t), wj⟩L2(Ω), j = 1, 2, ..., k, zk(0) = zk0 , z k t (0) = zk1 , u k(0) = uk0, u k t (0) = uk1, (27) where ⟨ , ⟩ is the inner product in L2(Ω) and zk0 = k∑ i=1 ⟨z0, wi⟩wi, u k 0 = k∑ i=1 ⟨u0, wi⟩wi, z k 1 = k∑ i=1 ⟨z1, wi⟩wi, u k 1 = k∑ i=1 ⟨u1, wi⟩wi. By the Projection Theorem in Hilbert spaces, the approximated initial data zk0 , u k 0, z k 1 , u k 1 are obtained via orthogonal projection onto the finite-dimensional subspace Vk, providing the best approximation in the corresponding norm. Since Vk is dense in H1 0 (Ω) and L2(Ω), and due to the finite-dimensional approximation property, the projections converge strongly to the original initial data. Furthermore, the Banach–Alaoglu theorem guarantees weak compactness of bounded sequences, but in this case, weak convergence implies the following strong convergence, zk0 → z0 and uk0 → u0 in H1 0 (Ω) and zk1 → z1 and uk1 → u1in L 2(Ω). (28) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 11 of 29 Based on standard existence theory for ordinal differential equations, the system (27) admits a unique local solution (zk, uk) on a maximal time interval [0, Tk), 0 < Tk < T, for each k ∈ N. Step 2. A priori Estimates: In this step, we show, by a priory estimates, that Tk = T, for each k ∈ N. We multiply the first equation by a′j(t) and the second equation by b′j(t) in (27), sum over j = 1, 2, ...k and add the two equations to obtain 1 2 d dt [ ρz∥zkt ∥22 + ρu∥ukt ∥22 + a1∥zkx∥22 + a3∥ukx∥22 + 2a2 ∫ Ω ukxz k xdx ] = −γ ∫ Ω |zkt (x, t)| p(.) dx− β ∫ Ω |ukt (x, t)| q(.) dx+ ∫ Ω ( zkt f(x, t) + ukt g(x, t) ) dx. (29) Integration of (29) over (0, t) leads to 1 2 ( ρz||zkt || 2 2 + ρu||ukt || 2 2 + a1||zkx|| 2 2 + a3||ukx|| 2 2 + 2a2 ∫ Ω ukxz k xdx ) + γ ∫ t 0 ∫ Ω |zkt (s)| p(.) dxds+ β ∫ t 0 ∫ Ω |ukt (s)| q(.) dxds = 1 2 ( ρz||zk1 || 2 2 + ρu||uk1|| 2 2 + ρz||zk0x|| 2 2 + ρu||uk0x|| 2 2 + 2a2 ∫ Ω zk0xu k 0xdx ) + ∫ t 0 ∫ Ω ( zkt f(x, t) + ukt g(x, t) ) dxds, for all t ≤ Tk. (30) Using the identity (26) and Young’s inequality on the last two terms, Eq. (30) becomes for any ε > 0, 1 2 ( ρz||zkt || 2 2 + ρu||ukt || 2 2 + C̃ ( ∥ux∥2 + ∥zx∥2 )) + γ ∫ Tk 0 ∫ Ω |zkt (s)| p(.) dxds+ β ∫ Tk 0 ∫ Ω |ukt (s)| q(.) dxds ≤ 1 2 ( ρz||zk1 || 2 2 + ρu||uk1|| 2 2 + ρz||zk0x|| 2 2 + ρu||uk0x|| 2 2 ) + c ( ||zk0x|| 2 2 + ||uk0x|| 2 2 ) + ε ∫ Tk 0 ( ρu ∥∥∥ukt ∥∥∥2 2 + ρz ∥∥∥zkt ∥∥∥2 2 ) ds+ Cε ∫ Tk 0 ∫ Ω ( |f(x, t)|2 + |g(x, t)|2 ) dxds. (31) Using (22) and recalling that f, g ∈ L2(Ω× (0, T )), we have zk0 −→ z0 and uk0 −→ u0 in H1 0 (Ω), zk1 −→ z1 and uk1 −→ u1 in L2(Ω) and appling Gronwall’s lemma, estimate (31) becomes, for some C > 0, C sup (0,Tk) [ ||zkt || 2 2 + ||ukt || 2 2 + ||zkx|| 2 2 + ||ukx|| 2 2 ] A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 12 of 29 + ∫ Tk 0 ∫ Ω ( γ ∣∣∣zkt (x, t)∣∣∣p(x) + β ∣∣∣ukt (x, t)∣∣∣q(x)) dxds ≤ Cε + εT sup (0,Tk) ( ρu ∥∥∥ukt ∥∥∥2 2 + ρz ∥∥∥zkt ∥∥∥2 2 ) ∀Tk ≤ T, k ≥ 1. Choosing ε = 1 4T , we find sup (0,Tk) [ ||zkt || 2 2 + ||ukt || 2 2 + ||zkx|| 2 2 + ||ukx|| 2 2 ] ≤ C. Therefore, the local solution (zk, uk) of system (27) can be extended to (0, T ), for all k ≥ 1. Furthermore, we have (zk), (uk) are bounded in L∞((0, T ), H1 0 (Ω)), (zkt ) is bounded in L∞((0, T ), L2(Ω)) ∩ Lp(.)(Ω× (0, T )), (ukt ) is bounded in L∞((0, T ), L2(Ω)) ∩ Lq(.)(Ω× (0, T )). Consequently, we have, up to two subsequences, zk → z and uk → u weakly * in L∞((0, T ), H1 0 (Ω)), zkt → zt weakly * in L∞((0, T ), L2(Ω)) and weakly in Lp(.)(Ω× (0, T )), ukt → ut weakly * in L∞((0, T ), L2(Ω)) and weakly in Lq(.)(Ω× (0, T )). Step 3. The Nonlinear terms: In this step, we show that | zkt |p(.)−2 zkt → | zt |p(.)−2 zt weakly in L p(.) p(.)−1 (Ω× (0, T )), | ukt |q(.)−2 ukt → | ut |q(.)−2 ut weakly in L q(.) q(.)−1 (Ω× (0, T )) and that (z, u) satisfies the partial differential equations of (Q) on Ω× (0, T ). Since (zkt ) is bounded in Lp(.)(Ω × (0, T )), then (|zkt | p(.)−2 zkt ) is bounded in L p(.) p(.)−1 (Ω × (0, T )). Hence, up to a subsequence, |zkt | p(.)−2 zkt⇀χ1 in L p(.) p(.)−1 (Ω× (0, T )). (32) Similarly, we have |ukt | q(.)−2 ukt⇀χ2 in L q(.) q(.)−1 (Ω× (0, T )). (33) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 13 of 29 We can show that χ1 = |zt|p(.)−2zt and χ2 = |ut|q(.)−2ut by following the same steps as in [34]. Now, integrate (27) on (0, t) to obtain ∀j < k,∫ Ω zkt wj(x)dx− ∫ Ω zk1wj(x)dx+ a1 ∫ t 0 ∫ Ω zkxwjx(x)dxds+ a2 ∫ t 0 ∫ Ω ukxwjx(x)dxds + γ ∫ t 0 ∫ Ω |zkt | p(.)−2 zkt wj(x)dxds = ∫ t 0 ∫ Ω wjf(x, t)dxds,∫ Ω uktwj(x)dx− ∫ Ω uk1wj(x)dx+ a3 ∫ t 0 ∫ Ω ukxwjx(x)dxds+ a2 ∫ t 0 ∫ Ω zkxwjx(x)dx + β ∫ t 0 ∫ Ω |ukt | q(.)−2 uktwj(x)dxds = ∫ t 0 ∫ Ω wjg(x, t)dxds. As k goes to +∞, we easily check that ∀j < k, ∫ Ω ztwj(x)dx− ∫ Ω z1wj(x)dx+ a1 ∫ t 0 ∫ Ω zxwjx(x)dxds+ a2 ∫ t 0 ∫ Ω uxwjx(x)dxds + γ ∫ t 0 ∫ Ω |zt|p(.)−2ztwj(x)dxds = ∫ t 0 ∫ Ω wjf(x, t)dxds,∫ Ω utwj(x)dx− ∫ Ω u1wj(x)dx+ a3 ∫ t 0 ∫ Ω uxwjx(x)dxds+ a2 ∫ t 0 ∫ Ω zxwjx(x)dx + β ∫ t 0 ∫ Ω |ut|q(.)−2utwj(x)dxds = ∫ t 0 ∫ Ω wjg(x, t)dxds. Consequently, we have ∀w ∈ H1 0 (Ω) ∫ Ω ztw(x)dx− ∫ Ω z1w(x)dx+ a1 ∫ t 0 ∫ Ω zxwx(x)dxds+ a2 ∫ t 0 ∫ Ω uxwx(x)dxds + γ ∫ t 0 ∫ Ω |zt|p(.)−2ztw(x)dxds = ∫ t 0 ∫ Ω wf(x, t)dxds,∫ Ω utw(x)dx− ∫ Ω u1w(x)dx+ a3 ∫ t 0 ∫ Ω uxwx(x)dxds+ a2 ∫ t 0 ∫ Ω zxwx(x)dxds + β ∫ t 0 ∫ Ω |ut|q(.)−2utw(x)dxds = ∫ t 0 ∫ Ω wg(x, t)dxds. All terms define absolute continuous functions, so we get, for a.e. t ∈ [0, T ] and ∀w ∈ A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 14 of 29 H1 0 (Ω),∫ Ω zttw(x)dx+ a1 ∫ Ω zxwx(x)dx+ a2 ∫ Ω uxwx(x)dx+ γ ∫ Ω |zt|p(.)−2ztw(x)dx = ∫ Ω wf(x, t)dx,∫ Ω uttw(x)dx+ a3 ∫ Ω uxwx(x)dx+ a2 ∫ Ω zxwx(x)dx+ β ∫ Ω |ut|q(.)−2utw(x)dx = ∫ Ω wg(x, t)dx. This implies that ρzztt − a1zxx − a2uxx + γ|zt|p(·)−2zt = f, in D′(Ω× (0, T )) ρuutt − a3uxx − a2zxx + β|ut|q(·)−2ut = g, in D′(Ω× (0, T )). This implies that (z, u) satisfies the two differential equations in (Q), on Ω× (0, T ). Step 4. The Initial Conditions: We can handle the initial conditions like the one in [34]. Hence, we deduce that (z, u) is the unique local solution of (Q). This completes the proof of Theorem 1. Now, we proceed to establish the local existence result for problem (P ), we first recall the following elementary inequalities:∣∣∣|a|k − |b|k ∣∣∣ ≤ C |a− b| ( |a|k−1 + |b|k−1 ) , (34) for some constant C > 0, all k ≥ 1 and all a, b ∈ R. Also∣∣∣|a|k0 a− |b|k0 b ∣∣∣ ≤ C |a− b| ( |a|k0 + |b|k0 ) , (35) for some constant C > 0, all k0 ≥ 0 and all a, b ∈ R. Remark 1. For a.e. x ∈ Ω and m(x) and ℓ(x) satisfying (A1), the functions h1(s) = c|s|m(x)−2 and h2(s) = d|s|ℓ(x)−2 are differentiable and |h′1(s)| = |c||m(x) − 1||s|m(x)−2, |h′2(s)| = |d||ℓ(x)− 1||s|ℓ(x)−2. Theorem 2. Let (u0, u1), (v0, v1) ∈ H1 0 (Ω) × L2(Ω) be given. Assume that (A1)-(A2) hold. Then, problem (P ) has a unique weak local solution (z, u) on [0, T ), in the sense of Definition 1, for some T > 0. Proof. EXISTENCE: Let v1, v2 ∈ L∞ ([0, T ), H1 0 (Ω) ) . Using Lemma 15 and the embedding property, we have ||h1(v1)||22 = ∫ Ω |v1|2(m(x)−1)dx ≤ |c| (∫ Ω |v1|2(m2−1)dx+ ∫ Ω |v1|2(m1−1)dx ) < +∞ ||h2(v2)||22 = ∫ Ω |v2|2(ℓ(x)−1)dx ≤ |d| (∫ Ω |v2|2(ℓ2−1)dx+ ∫ Ω |v2|2(ℓ1−1)dx ) < +∞. (36) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 15 of 29 Hence, h1(v1), h2(v2) ∈ L∞([0, T ), L2(Ω)) ⊂ L2(Ω× (0, T )). Therefore, for each v1, v2 ∈ L∞([0, T ), H1 0 (Ω)), there exists a unique solution (z, u) ∈ L∞([0, T ), H1 0 (Ω)), zt ∈ L∞([0, T ), L2(Ω)) ∩ Lp(Ω× (0, T )) and ut ∈ L∞([0, T ), L2(Ω)) ∩ Lq(Ω× (0, T )) satisfying the following nonlinear problem ρzztt − a1zxx − a2uxx + γ |zt|p(x)−2 zt = h1(v1) in Ω× (0, T ) , ρuutt − a3uxx − a2zxx + β |ut|q(x)−2 ut = h2(v2) in Ω× (0, T ) , u = z = 0 on ∂Ω× (0, T ) , u (0) = u0 and ut (0) = u1 in Ω, z (0) = z0 and zt (0) = z1 in Ω. (R) Now, let WT = { w ∈ L∞((0, T ), H1 0 (Ω))/wt ∈ L∞((0, T ), L2(Ω)) } , and define the map K :WT ×WT −→WT ×WT by K(v1, v2) = (z, u). We note that WT is a Banach space with respect to the following norm ||w||2WT = sup (0,T ) ∫ Ω |wx|2dx+ sup (0,T ) ∫ Ω |wt|2dx, and K is well defined by virtue of Theorem 1. In what follows, we prove that K is a contraction mapping from a closed bounded ball B(0,M) into itself, where B(0,M) = { (v1, v2) ∈WT ×WT / ∥(v1, v2)∥WT×WT ≤M } , for M > 1 and T0 > 0 to be fixed later. Multiplying the first equation in (R) by zt, the second one by ut and integrating the two results over Ω× (0, t) we get, for all t ≤ T, ρu 2 ∥ut∥22 + ρz 2 ∥zt∥22 + a3 2 ∥ux∥22 + a1 2 ∥zx∥22 + a2 ∫ Ω uxzxdx− ρu 2 ∥u1∥22 − ρz 2 ∥z1∥22 − ρu 2 ∥u0x∥22 − ρz 2 ∥z0x∥22 − a2 ∫ Ω u0xz0xdx+ γ ∫ t 0 ∫ Ω |zt|p(x) dxds+ β ∫ t 0 ∫ Ω |ut|q(x) dxds = ∫ t 0 ∫ Ω zth1(v1)dxds+ ∫ t 0 ∫ Ω uth2(v2)dxds. (37) Using the definitions of hi, i = 1, 2, Lemma 15 and Young’s and Poincaré’s inequalities, we have for ε > 0,∫ Ω |v1|m(x)−2v1ztdx ≤ ερz 4 ∫ Ω z2t dx+ C ε ∫ Ω |v1|2(m(x)−1)dx ≤ ερZ 4 ∫ Ω z2t dx+ Ce ε [ ||v1x||2m2−2 2 + ||v1x||2m1−2 2 ] . (38) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 16 of 29 Similarly, we have∫ Ω |v2|ℓ(x)−2v2utdx ≤ ερu 4 ∫ Ω u2tdx+ C ε ∫ Ω |v2|2(ℓ(x)−1)dx ≤ ερu 4 ∫ Ω u2tdx+ Ce ε [ ||v2x||2ℓ2−2 2 + ||v2x||2ℓ1−2 2 ] . (39) Thus, (37) becomes ρu 2 ∥ut∥22 + ρz 2 ∥zt∥22 + a3 2 ∥ux∥22 + a1 2 ∥zx∥22 + a2 ∫ Ω uxzxdx ≤ λ0 + εTρz 4 sup (0,T ) ∫ Ω z2t dx+ εTρu 4 sup (0,T ) ∫ Ω u2tdx + Ce ε [ ||v1x||2m1−2 2 + ||v1x||2m2−2 2 + ||v2x||2ℓ2−2 2 + ||v2x||2ℓ1−2 2 ] , where, by using (26), λ0 = ρu 2 ∥u1∥22 + ρz 2 ∥z1∥22 + ρu 2 ∥u0x∥22 + ρz 2 ∥z0x∥22 + a2 ∫ Ω u0xz0xdx ≥ 0, and Ce is the embedding constant. Choosing ε such that εT = 1, we get ||z||2WT + ||u||2WT ≤ λ0 + TC ( ||v1||2(m1−1) WT + ||v1||2(m2−1) WT + ||v2||2(ℓ1−1) WT + ||v2||2(ℓ2−1) WT ) . Suppose that max {||v1||WT , ||v2||WT } ≤M , for some M large. Then, we have for large M > 0, ||u||2WT + ||z||2WT ≤ λ0 + TCM̃ ≤M2, where M̃ = max{M2(m2−1),M2(ℓ2−1)} and M2 > λ0 and T ≤ T0 < M2−λ0 CM̃ . Hence, we conclude that that K maps B(0,M) into B(0,M). Next, we prove, for T0(even smaller), K is a contraction. For this purpose, let (z1, u1) = K(v1, ṽ1) and (z2, u2) = K(v2, ṽ2) and set (Z,U) = (v1 − v2, ṽ1 − ṽ2) then (Z,U) satisfies the following ρzZtt − a1Zxx − a2Uxx + γ ( |v1t|p(·)−2v1t − |v2t|p(·)−2v2t ) = c ( |v1|m(·)−2v1 − |v2|m(·)−2v2 ) , ρuUtt − a3Uxx − a2Zxx + β ( |ṽ1t|q(·)−2ṽ1t − |ṽ2t|q(·)−2ṽ2t ) = d ( |ṽ1|ℓ(·)−2ṽ1 − |ṽ2|ℓ(·)−2ṽ2 ) , U(x, 0) = U0(x), Ut(x, 0) = U1(x), Z(x, 0) = Z0(x), Zt(x, 0) = Z1(x), Z(0, t) = Z(1, t) = U(0, t) = U(1, t) = 0. (40) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 17 of 29 Multiplication the first equation by Zt and the second by Ut, integration over Ω × (0, t) and addition of the two equations yield ρu 2 ∥Ut∥22 + ρz 2 ∥Zt∥22 + a3 2 ∥Ux∥22 + a1 2 ∥Zx∥22 + a2 ∫ Ω UxZxdx + γ ∫ t 0 ∫ Ω ( |v1t|p(·)−2v1t − |v2t|p(·)−2v2t ) Ztdxds+ β ∫ t 0 ∫ Ω ( |ṽ1t|q(·)−2ṽ1t − |ṽ2t|q(·)−2ṽ2t ) Utdxds = ∫ t 0 ∫ Ω (h1(v1)− h1(v2))Ztdxds+ ∫ t 0 ∫ Ω (h2(ṽ1)− h2(ṽ2))Utdxds. Hence, we have ρu 2 ∥Ut∥22 + ρz 2 ∥Zt∥22 + a3 2 ∥Ux∥22 + a1 2 ∥Zx∥22 + a2 ∫ Ω UxZxdx ≤ ∫ t 0 ∫ Ω (h1(v2)− h1(v2))Ztdxds+ ∫ t 0 ∫ Ω (h2(ṽ1)− h2(ṽ2))Utdxds. (41) Now, we evaluate I1 = ∫ Ω |h1(v1)− h1(v2)||Zt| and I2 = ∫ Ω |h2(ṽ1)− h2(ṽ2)||Ut|. Therefore, I1 = ∫ Ω |h1(v1) − h1(v2)||Zt| = ∫ Ω |h′1(ξ)||v||Zt|,where v = v1 − v2 and ξ = αv1 − (1− α)v2, 0 ≤ α ≤ 1. Applying Young’s inequality, we get for any δ > 0 and some positive constant C, I1 ≤ δ 2 ∫ Ω Z2 t dx+ 2 δ ∫ Ω |h′1(ξ)|2|v|2dx ≤ δ 2 ∫ Ω Z2 t dx+ C δ ∫ Ω |αv1 − (1− α)v2|2(m(x)−2)|v|2dx ≤ δ 2 ∫ Ω Z2 t dx + Cδ (∫ Ω |v| 2n n−2 )n−2 n × [(∫ Ω |αv1 + (1− α)v2|n(m2−2) ) 2 n + (∫ Ω |αv1 + (1− α)v2|n(m1−2) ) 2 n ] . By recalling Lemma (15), we arrive at I1 ≤ δ 2 ∫ Ω Z2 t dx+ CδCe∥vx∥22 ( ∥v1x∥ 2(m2−2) 2 + ∥v1x∥ 2(m1−2) 2 + ∥v2x∥ 2(m2−2) 2 + ∥v2x∥ 2(m1−2) 2 ) ≤ δ 2 ∫ Ω Z2 t dx+ CδCeM̃∥vx∥22. Similarly, we can show that I2 ≤ δ 2 ∫ Ω U2 t dx+ CδCeM̃∥ṽx∥22, where ṽ = ṽ1 − ṽ2. Therefore, (41) takes the form ∥(Z,U)∥2WT×WT ≤ δ 2 T0C∥(Z,U)∥2WT×WT + 4CδM̃T0C∥(v, ṽ)∥2WT×WT . A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 18 of 29 Choosing δ small enough, we arrive at ∥(Z,U)∥2WT×WT ≤ CT0∥(v, ṽ)∥2WT×WT . Taking T0 small enough, we get, for some 0 < k < 1, ∥(Z,U)∥WT×WT ≤ k∥(v, ṽ)∥2WT×WT . Thus K is a contraction. The Banach fixed theorem implies the existence of a unique (z, u) ∈ D(0,M), such that K(z, u) = (z, u). Hence,, (z, u) is a weak solution of system (6). The uniqueness of this solution can be obtained by applying the energy method. 5. Global existence In this section, we prove that system (6) has a global solution if p ≤ m and q ≤ ℓ. Proposition 1. Assume that p ≤ m and q ≤ ℓ. Then, system 6 admits a unique global solution. Proof. Similar to [25], we define the following E(t) := E(t) + 2c ∫ Ω |z|m(x) m(x) dx+ 2d ∫ Ω |u|ℓ(x) ℓ(x) dx = 1 2 ∫ Ω [ ρzz 2 t + ρuu 2 t + a3u 2 x + a1z 2 x + 2a2zxux ] dx +c ∫ Ω |z|m(x) m(x) dx+ d ∫ Ω |u|ℓ(x) ℓ(x) dx. (42) Therefore, E ′(t) = −γ ∫ Ω |zt|p(·)dx− β ∫ Ω |ut|q(·)dx+ 2c ∫ Ω |z|m(·)−2ztdx+ 2d ∫ Ω |u|ℓ(·)−2utdx. By using Young’s inequality, we obtain for any ε, δ > 0, E ′(t) ≤ −γ ∫ Ω |zt|p(·)dx− β ∫ Ω |ut|q(·)dx (43) +ε ∫ Ω |zt|m(.)dx+ δ ∫ Ω |ut|ℓ(·)dx (44) + ∫ Ω Cε(x)|z|m(.)dx+ ∫ Ω Cδ(x)|u|ℓ(·)dx. (45) By noting that p ≤ m and q ≤ ℓ, we have E ′(t) ≤ −γ ∫ Ω |zt|p(·)dx− β ∫ Ω |ut|q(·)dx+ Cε ∫ Ω |zt|p(·)dx+ Cδ ∫ Ω |ut|q(·)dx (46) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 19 of 29 + ∫ Ω Cε(x)|z|m(·)dx+ ∫ Ω Cδ(x)|u|ℓ(·)dx. (47) Choosing ε and δ such that γ − Cε > 0 and β − Cδ > 0, we obtain E ′(t) ≤ C ( m2 ∫ Ω |z|m(x) m(x) dx+ ℓ2 ∫ Ω |u|ℓ(·) ℓ(x) dx ) ≤ CE(t). (48) A simple integration gives E(t) ≤ E(0)eCt. (49) The last estimate together with the continuation principle completes our proof. 6. Blow-up In this section, we show that the solution of system (6) blows up in a finite time. Our blow-up result reads as follows: Theorem 3. Assume that (A1) and (A2) hold and E(0) < 0. Then the solution of system 6 blows-up in a finite time. Proof. We set H(t) := −E(t) then H ′(t) = −E′(t) ≥ 0 and then for every t ∈ [0, T ), we have 0 < H(0) ≤ H(t) ≤ c m1 ∫ Ω |z|m(x)dx+ d ℓ1 ∫ Ω |u|ℓ(x)dx. (50) We then define F (t) = H1−α(t) + ε ∫ Ω (ρzzzt + ρuuut)dx, (51) for 0 < α < 1 and a positive number ε to be chosen later. By taking the derivative of F and using Eq. (6), we obtain F ′(t) = (1− α)H−α(t)H ′(t) + ε ∫ Ω ρzz 2 t dx+ ε ∫ Ω ρuu 2 tdx− εa3 ∫ Ω u2xdx− εa1 ∫ Ω z2xdx −2εa2 ∫ Ω uxzxdx− εγ ∫ Ω z|zt|p(·)−2ztdx− εβ ∫ Ω u|ut|q(·)−2utdx +εc ∫ Ω |z|m(·)dx+ εc ∫ Ω |u|ℓ(·)dx. (52) Adding and subtracting ε(1 − θ)m1ℓ1H(t), for 0 < θ < 1, to the right-hand side of (52), we arrive at F ′(t) = (1− α)H−α(t)H ′(t) + ε(1− θ)m1ℓ1H(t) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 20 of 29 +ερz ( 1 + (1− θ)m1ℓ1 2 )∫ Ω z2t dx+ ερu ( 1 + (1− θ)m1ℓ1 2 )∫ Ω u2tdx −εa3 ( (1− θ)m1ℓ1 2 − 1 )∫ Ω u2xdx− εa1 ( (1− θ)m1ℓ1 2 − 1 )∫ Ω z2xdx −2εa2 ( (1− θ)m1ℓ1 2 − 1 )∫ Ω uxzxdx+ εcℓ1 ∫ Ω |z|m(·)dx+ εcm1 ∫ Ω |u|ℓ(·)dx −εγ ∫ Ω z|zt|p(·)−2ztdx− εβ ∫ Ω u|ut|q(·)−2utdx. (53) For θ small enough, we have for some positive constant η F ′(t) ≥ (1− α)H−α(t)H ′(t) +εη [ H(t) + ||ut||22 + ||zt||22 + ||ux||22 + ||zx||22 + ∫ Ω |z|m(·)dx+ ∫ Ω 1|u|ℓ(·)dx ] −εγ ∫ Ω z|zt|p(·)−2ztdx− εβ ∫ Ω u|ut|q(·)−2utdx, (54) Now, by using Young’s inequality, we estimate the last term in (54) as follows: For any σ1, σ2 > 0, we have ∫ Ω |z||zt|p(·)−1dx ≤ 1 p1 ∫ Ω σ p(x) 1 |z|p(x)dx+ (p2 − 1) p2 ∫ Ω σ −p(x) p(x)−1 1 |zt|p(x)dx,∫ Ω |u||ut|q(·)−1dx ≤ 1 q1 ∫ Ω σ q(x) 2 |u|q(x)dx+ (q2 − 1) q2 ∫ Ω σ −q(x) q(x)−1 2 |ut|q(x)dx. (55) Therefore, by choosing σ1 and σ2 such that σ −p(x) p(x)−1 1 = ξ1H −α(t), σ −p(x) p(x)−1 2 = ξ2H −α(t) (56) for sufficiently large constants ξi, i = 1, 2, to be specified later, and substituting these into (55), we obtain: ∫ Ω |z||zt|p(·)−1dx+ ∫ Ω |u||ut|q(·)−1dx ≤ 1 p1 ∫ Ω ξ 1−p(x) 1 |z|p(x)Hα(p(x)−1)dx+ (p2 − 1) cp2 ξ1H −α(t)H ′(t) + 1 q1 ∫ Ω ξ 1−q(x) 2 |u|q(x)Hα(q(x)−1)dx+ (q2 − 1) dq2 ξ2H −α(t)H ′(t). (57) Combining (54) and (57), we obtain F ′(t) ≥ [ (1− α)− ε ( (p2 − 1) p2 ξ1 ) − ε ( (q2 − 1) q2 ξ2 )] H−α(t)H ′(t) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 21 of 29 +εη [ H(t) + ||ut||22 + ||zt||22 + ||ux||22 + ||zx||22 + ∫ Ω |z|m(·)dx+ ∫ Ω |u|ℓ(·)dx ] −cεξ 1−p1 1 p1 Hα(p2(x)−1)(t) ∫ Ω |z|p(x)dx− dε ξ1−q1 2 q1 Hα(q2(x)−1)(t) ∫ Ω |u|q(x)dx.(58) Recalling (16) and (17), then Eq. (18) becomes Hα(p2(x)−1)(t) ∫ Ω |z|p(x)dx+Hα(q2(x)−1)(t) ∫ Ω |u|q(x)dx ≤ Hα(p∗(x)−1)(t) [∫ Ω |z|p(x)dx+ ∫ Ω |u|q(x)dx ] ≤ C ( ϱ(z) p1 m1 +α(p∗(x)−1) + ϱ(z) p2 m1 +α(p∗(x)−1) + ϱ(u) q1 ℓ1 +α(p∗(x)−1) + ϱ(u) q2 ℓ1 +α(p∗(x)−1) ) ≤ C ( ||zx||22 + ϱ(z) + ||ux||22 + ϱ(u) ) , where p∗(x) = max{p2(x), q2(x)}. Using (58), we arrive at F ′(t) ≥ [ (1− α)− ε ( (p2 − 1) p2 ξ1 ) − ε ( (q2 − 1) q2 ξ2 )] H−α(t)H ′(t) +ε(η − λ) [ H(t) + ||ut||22 + ||zt||22 + ||ux||22 + ||zx||22 + ϱ(z) + ϱ(u) ] , (59) λ := c ξ 1−p1 p1 + d ξ1−q1 q1 > 0 and ξ = max{ξ1, ξ2}. Now, we choose ξ large enough such that µ := η − λ > 0. Then, we pick ε small enough so that (1− α)− ε [( (p2 − 1) p2 ξ1 ) + ( (q2 − 1) q2 ξ2 )] ≥ 0, and F (0) = H1−α(0) + ε ∫ Ω (ρzz0z1 + ρuu0u1)dx > 0. Using the last results, recalling (17) and (16), Eq. (60) becomes F ′(t) ≥ µε [ H(t) + ||ut||22 + ||zt||22 + ||ux||22 + ||zx||22 + ϱ(z) + ϱ(u) ] ≥ µε [ H(t) + ||ut||22 + ||zt||22 + ||ux||22 + ||zx||22 + ||z||m1 m1 + ||u||ℓ1ℓ1 ] > 0. (60) Hence, F is non-decreasing; that is, we have F (t) ≥ F (0), ∀ t ≥ 0. (61) A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 22 of 29 Using (51) and (60), we have F ′(t) ≥ νF 1 1−α , ∀ t ≥ 0. (62) A simple integration of (62), we obtain F α 1−α (t) ≥ 1 F −α 1−α (0)− νtα 1−α , (63) where 0 < α < 1 and ν > 0. Therefore, (63) shows that F blows-up in the finite time T∗ ≤ 1− α να [F (0)] α 1−α . (64) This completes the proof. Remark 2. In the context of swelling porous-elastic systems, blow-up represents the on- set of mechanical failure, which can manifest physically as excessive deformation, material rupture, or cracking, depending on the specific application. When a solution blows up in finite time, it signifies that certain physical quantities, such as stress, strain, or displace- ment, become unbounded, indicating an irreversible breakdown of the material structure. Thus, blow-up in our model provides a mathematical framework for predicting critical thresholds beyond which the material loses stability, aiding in failure analysis and design optimization of porous-elastic structures. 7. Numerical Tests In the following section, we illustrate the blow up results proved in Theorems 6.1. We perform four numerical tests for the blow up behavior of a one-dimensional real-valued function. We discretize the system 1.6 using a second order finite difference method explicit in time and in space. For more stability, we combine the finite difference method with the conservative scheme of Lax-Wendroff. For more details, we refer to our previous works [12, 35]. We examine the following four tests. For these test, we define the parameters a1 = a3 = 1 and a2 = 0.95 satisfying the conditions (A2). The used spatial-temporal domain is [0, 1]2 × [0, 1]: • TEST 1: In the first test, we examine the blow up of the energy function using the nonlinear exponent function p(x) = q(x) = 1 + 1 (1 + x2) < m(x) = ℓ(x) = 2 + 2 (1 + x2) , which satisfies the condition (A1). A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 23 of 29 • TEST 2: In the second numerical test, we modify the inequality in the first test p(x) = q(x) = m(x) = 1 + 1 (1 + x2) < ℓ(x) = 2 + 2 (1 + x2) , which satisfies the condition (A1). • TEST 3: Similarly, in the third numerical test, we use p(x) = q(x) = ℓ(x) = 1 + 1 (1 + x2) < m(x) = 2 + 2 (1 + x2) , which satisfies the condition (A1). • TEST 4: In the fourth numerical test, we set the following equality of the non-linear exponent functions p(x) = q(x) = m(x) = ℓ(x) = 1 + 1 (1 + x2) , which satisfies the condition (A1). We run our code using the following initial solution: u(x, 0) = sin(πx), (65) z(x, 0) = 2x(x− 1) (66) As mentioned in the system (1.6), we set u1(x, 0) = u(x, , 0). We also have to mention that to visualize the blow up behavior of the initial solution (65), we use a very small and constant temporal step ∆t = 10−5 and the equidistant spatial step ∆x = ∆y = 10−2. In the left two column of the Figures 3-6, we plot the cross section of the evolution of the solution at different time steps t = 0, t = 0.25 and t = 0.75. In the right column of the Figures 3-6, we plot the blow up of the energies for the four tests. A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 24 of 29 0 0.002 0.004 0.006 0.008 0.01 0.012 -50 0 50 u (t , 0. 2 5) 0 0.002 0.004 0.006 0.008 0.01 0.012 -20 0 20 u (t , 0 .5 0) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time -5 0 5 u (t , 0. 75 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 -5 0 5 z (t , 0. 25 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 -5 0 5 z (t , 0. 50 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time -1 0 1 z (t , 0. 75 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time 0 500 1000 1500 2000 2500 3000 E (t ) Figure 3: Test 1: Blow up under p(x) < m(x) and ℓ(x) < m(x) . 0 0.002 0.004 0.006 0.008 0.01 0.012 -50 0 50 u (t , 0. 2 5) 0 0.002 0.004 0.006 0.008 0.01 0.012 -20 0 20 u (t , 0. 50 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time -5 0 5 u (t , 0. 75 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 -5 0 5 z (t , 0. 25 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 -5 0 5 z (t , 0. 50 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time -1 0 1 z (t , 0. 75 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time 0 500 1000 1500 2000 2500 3000 E (t ) Figure 4: Test 2: Blow up under p(x) = m(x) and ℓ(x) < m(x) . A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 25 of 29 0 0.002 0.004 0.006 0.008 0.01 0.012 -50 0 50 u (t , 0. 2 5) 0 0.002 0.004 0.006 0.008 0.01 0.012 -20 0 20 u (t , 0 .5 0) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time -5 0 5 u (t , 0. 75 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 -5 0 5 z (t , 0. 25 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 -5 0 5 z (t , 0. 50 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time -1 0 1 z (t , 0. 75 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time 0 500 1000 1500 2000 2500 3000 E (t ) Figure 5: Test 3: Blow up under p(x) < m(x) and ℓ(x) = m(x) . 0 0.002 0.004 0.006 0.008 0.01 0.012 -50 0 50 u (t , 0. 2 5) 0 0.002 0.004 0.006 0.008 0.01 0.012 -20 0 20 u (t , 0. 50 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time -5 0 5 u (t , 0. 75 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 -5 0 5 z (t , 0. 25 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 -5 0 5 z (t , 0. 50 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time -1 0 1 z (t , 0. 75 ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time 0 500 1000 1500 2000 2500 3000 E (t ) Figure 6: Test 4: Blow up under p(x) = m(x) and ℓ(x) = m(x) . Finally, we remarked that even for a fine spatial and temporal discretization, the blow up occurs after a finite number of steps for all the Tests. Conclusion In this work, we investigated a swelling soil system incorporating two nonlinear damp- ing and source terms of variable exponent-type. By employing the Faedo-Galerkin method and the Banach Contraction Theorem, we established the local existence and uniqueness A. M. Al-Mahdi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6073 26 of 29 of weak solutions under suitable conditions on the variable exponent functions. Further- more, we demonstrated the global existence of solutions and identified conditions leading to finite-time blow-up. A key contribution of this study is the consideration of damp- ing terms with variable exponents, which significantly generalizes classical models with constant exponent damping. This formulation allows for a more flexible and realistic representation of energy dissipation, capturing heterogeneous material properties and dy- namic changes in the system. The presence of variable exponent damping plays a crucial role in influencing the stability and long-term behavior of solutions. Finally, we provided numerical simulations to illustrate the blow-up behavior, further validating our theoretical findings. While this study establishes significant results on the existence, uniqueness, and blow-up of solutions for swelling porous-elastic systems with variable exponent damping and source terms, several questions remain open for further investigation: I. Extension to Higher Dimensions and General Domains The current analysis is restricted to one-dimensional settings. Extending the results to higher-dimensional porous-elastic systems with variable exponent nonlinearity is a significant challenge that could lead to new theoretical developments. II. 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