Abstract
In the present study, we consider a thermal-Timoshenko-beam system with suspenders and Kelvin–Voigt damping type, where the heat is given by Cattaneo's law. Using the existing semi-group theory and energy method, we establish the existence and uniqueness of weak global solution, and an exponential stability result. The results are obtained without imposing the equal-wave speed of propagation condition.
2010 MSC: 35D30, 35D35, 35B35.
1. Problem setting and introduction
In the present study, we consider a cable-suspended beam structure such as the suspension bridge, where the roadbed has a negligible sectional dimension in comparison with its length (span of the bridge). Therefore, it is modeled in Timoshenko theory through a one-dimensional extensible beam, while the (main) suspension cable models an elastic string that is coupled to the deck. The equations of motion describing such Timoshenko-suspended-beam system, see [1–9], are given by where u = u(x, t) is the vertical displacement of the vibrating spring of main cable, φ = φ(x, t) is the transverse displacement, ψ = ψ(x, t) is the rotation angle, and L, ρ, A, and I are, respectively, length, mass density, cross-section area, and moment of inertia. The constitutive laws V, Q, S, and M are defined by where the physical parameters α, λ, E, G, and k are, respectively, the elastic modulus of the string, the stiffness of elastic springs, the Young's modulus of the beam, the shear modulus, and the shear correction coefficient of the beam. Generally, the system (1.1) is not exponentially stable, see for instance [10, 11], and the references therein. Therefore, we need to introduce a dissipative mechanism to achieve an exponential stability. A common and powerful way of stabilizing hyperbolic systems from mechanical structures in literature is through thermal damping, see [12], where a generalized theory on thermoelasticity is established. Assuming the cable is thermally insulated and consider a stress–strain constitutive law of Kelvin–Voigt type, see [11], then (1.2) takes the form where γ0 and γ1 are damping coefficients, θ = θ(x, t) is the temperature difference, and β > 0 is a coupling constant. When the heat conduction θ in (1.3) is governed by Cattaneo's law [13–15], we have the following: where q = q(x, t) is the heat flux and ρ3, τ, and σ > 0 are coupling constants. Considering linear damping force with damping coefficient γ0 on the vertical displacement of suspenders and by setting L = 1, ρ1 = ρA, ρ2 = ρI, k1 = kGA, and k2 = EI, then substituting (1.3) into (1.1) and coupling it with (1.4), we arrive at the following system: We supplement system (1.5) with the boundary conditions as follows: and the initial data are The main focus of this article was to investigate system (1.5)−(1.7). We establish the well-posedness and the asymptotic behavior of solution by using the semi-group and the multiplier methods. For related results to system (1.5)−(1.7), we mention the result of Bochicchio et al. [16], where the authors considered system (1.5) with heat conduction governed by Fourier 's law (τ = 0), γ1 = γ2 = 0, and linear frictional damping on (1.5)1 and (1.5)2. They proved an exponential stability result and numerical analysis of the system. Very recently, Mukiawa et al. [17] studied (1.5) with general, delay, and weak internal damping on the first equation and established a general stability result. We also mention the study of Enyi [18], the author proved exponential stability results for thermoelastic Timoshenko beam systems with full and partial Kelvin–Voigt damping, where the heat conduction is governed by the Cattaneo law of heat transfer. There are many closely related Timoshenko systems in literature, which have discussed a lack of exponential stability, see [11, 19, 20], and the references therein. In comparison to the present system, there is no ambiguity since the system is fully damped. Another interesting direction that can be considered is a type of thermoelastic system governed by Saint-Venant's principle on bounded bodies, see [21], where the decay estimates of two-temperature model are obtained. For more related results, the reader should consult the following articles [22–26] and the references therein. The rest of this study is organized as follows: In Section 2, we prove an existence and uniqueness result. In Section 3, we state and prove the main stability result.
2. Well-posedness
In this section, we transform system (1.5)−(1.7) into semi-group setting and establish the existence and uniqueness result. Let 〈., .〉 and ||.|| denote, respectively, the inner product and the norm in L2(0, 1).
We shall convert Problem (1.5)–(1.7) into the Cauchy form
Define appropriate spaces and use the semi-group method, see Pazy [27], to establish the well-posedness.
To this end, we set W = (u, v, φ, w, ψ, z, θ, q)T, where v = ut, w = φt, and z = ψt. Thus, problem (1.5)–(1.7) becomes where and Next, we define the Sobolev spaces as follows: The phase space of our problem is the following Hilbert space, We endow with the following inner product: for any , and norm for any W = (u, v, φ, w, ψ, z, θ, q)T ∈ .
The domain of is defined as,
Lemma 2.1. The operator is monotone.
Proof. Let W = (u, v, φ, w, ψ, z, θ, q) ∈ , then using integration by parts and the boundary conditions (1.6), we get,
Therefore, is monotone. □
Lemma 2.2. The operator is maximal.
Proof. Let F = (f1, f2, f3, f4, f5, f6, f7, f8)T ∈ . We consider the stationary problem where W = (u, v, φ, w, ψ, z, θ, q). Now, from (2.3), we get, From (2.4)1, (2.4)3, and (2.4)5, we have v = u − f1, w = φ − f3, and z = ψ − f5, respectively. Therefore, (2.4) becomes, We define the following bilinear form on ℍ × ℍ and linear form on ℍ, where , as follows:
and
for every (u, φ, ψ, θ, q), (u*, φ*, ψ*, θ*, q*) ∈ ℍ.
When ℍ is endowed with the following norm, it is easy to see that is a continuous and coercive bilinear form on ℍ × ℍ, and is a linear continuous form on ℍ. Therefore, by the Lax–Milgram theorem, there exists a unique (u, φ, ψ, θ, q) ∈ ℍ such that It follows from (2.4)1, (2.4)3, and (2.4)5 that , , and , respectively. Then, using regularity theory, it follows from (2.5)1, (2.5)2, and (2.5)3, that u, φ, ψ ∈ H2(0, 1). Moreover, from (2.5)4 and (2.5)5, we deduce that and . Therefore, and satisfies (2.3), that is, is maximal.
□
On account of Lemma 2.1 and Lemma 2.2, we apply the semi-group theory for linear operator, see [27], and immediately have the following result.
Theorem 2.1. Let W0 ∈ be given, then the Cauchy Problem (2.2) has a unique local weak solution,
Remark 2.1. One can easily compute [see (3.3)] that the solution of (1.5)–(1.7) is given by Theorem 2.1 that satisfies Thus, the solution W is global, that is, if W0 ∈ then .
Now, due to the density of in , we can announce the following result.
Theorem 2.2. Given , then problem (1.5)–(1.7) has a unique global solution in the class
3. Stability result
This section is devoted to the exponential stability of system (1.5)–(1.7). The energy functional associated with problem (1.5)−(1.7) is defined by The main stability result is as follows:
Theorem 3.1. The energy functional defined in (3.1) decays exponentially as time approaches infinity. That is, there exist two constants K, δ > 0 such that
3.1. Proof of Theorem 3.1
We provide several Lemmas to facilitate the proof of Theorem (3.1).
Lemma 3.1. Let (φ, ψ, θ, q) be the solution of (1.5). Then, the energy functional (3.1) satisfies Proof. Multiplying (1.5)1 by ut, (1.5)2 by φt, (1.5)3 by ψt, (1.5)4 by θ, (1.5)5 by q, integrating over (0, 1), using integration by parts and the boundary conditions (1.6), we have, and Adding (3.4)–(3.8), we obtain, The computations above are done for regular solution. However, the result remains true for weak solution by density argument. □
Remark 3.1. The lemma above implies that the energy (3.1) is decreasing and bounded above by E(0).
Now, we construct a suitable Lyapunov functional L such that for some a1, a2 > 0, and show that L satisfies for some η > 0 from which, we obtain for some ϖ > 0. The exponential decay of L in (3.12) will then imply the exponential decay of the energy functional To achieve (3.10)–(3.12), we define L as follows: for some N, N1, N2, N3 > 0 to be specified later, and Let us mention that routine computations, applying Young's, Cauchy–Schwarz, and Poincaré's inequalities give (3.10). Next, we provide some Lemmas needed to establish (3.11)–(3.12).
Lemma 3.2. The functional G1, along the solution of system (1.5)−(1.7) satisfies the estimate Proof. Differentiating G1, using (1.5)1, (1.5)2, and (1.5)3, then applying integration by parts and the boundary conditions (1.6), we obtain Exploiting Young's and Poincaré's inequalities, we obtain, By setting , and , we obtain (3.15). □
Lemma 3.3. The functional G2, along the solution of system (1.5)− (1.7), satisfies the estimate Proof. Differentiation of G2, using (1.5)3 and (1.5)4, and applying integration by parts leads to where Using Cauchy–Schwarz, we note that It follows by Young's and Cauchy–Schwarz inequalities that Hence, we obtain (3.18), with and . □
Lemma 3.4. The functional G3, along the solution of (1.5), satisfies, the estimate Proof. Differentiation of G3, using (1.5)2 and (1.5)4, integration by parts and boundary conditions, we get, where, Exploiting Cauchy–Schwarz inequality, we see that Therefore, using Young's, Cauchy–Schwarz, and Poincaré's inequalities, we get Thus, taking , , and , we obtain (3.20). □
Now, we give the proof of Theorem 3.1.
Proof. Using Lemma 3.1 and Lemmas 3.2−3.4, it follows from (3.13) that By setting we obtain Now, we choose N3 large so that Next, we select N2 large enough such that Lastly, we choose N very large enough so that (3.10) remain valid and Thus, we have for some η > 0. Using (3.1) and Poincaré's inequality, we get for some positive constant η1. Integrating (3.24) over (0, t) yields for some ϖ > 0 Hence, the exponential estimate of the energy functional in (3.2) follows from (3.25) and the equivalent relation (3.10). □
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
This study was funded by the Institutional Fund Projects # IFP-A-2022-2-1-04.
Acknowledgments
The authors gratefully acknowledge the technical and financial support from the Ministry of Education and the University of Hafr Al Batin, Saudi Arabia.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
Timoshenko, thermoelasticity, suspenders, Cattaneo's law, Kelvin–Voigt, well-posedness, stability
Citation
Mukiawa SE, Khan Y, Al Sulaimani H, Omaba ME and Enyi CD (2023) Thermal Timoshenko beam system with suspenders and Kelvin–Voigt damping. Front. Appl. Math. Stat. 9:1153071. doi: 10.3389/fams.2023.1153071
Received
28 January 2023
Accepted
24 April 2023
Published
19 May 2023
Volume
9 - 2023
Edited by
Ibrahim A. Abbas, Sohag University, Egypt
Reviewed by
Emad Awad, Alexandria University, Egypt; Carlos Alberto Nonato, Federal University of Bahia (UFBA), Brazil; Carlos Alberto Raposo Da Cunha, Federal University of Bahia (UFBA), Brazil
Updates
Copyright
© 2023 Mukiawa, Khan, Al Sulaimani, Omaba and Enyi.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Soh Edwin Mukiawa mukiawa@uhb.edu.sa
†ORCID: Soh Edwin Mukiawa orcid.org/0000-0002-6668-1107
McSylvester Ejighikeme Omaba orcid.org/0000-0002-5163-229X
Cyril Dennis Enyi orcid.org/0000-0001-9658-5864
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.