On the double Moore–Gibson–Thompson system of thermoviscoelasticity

Studies in Applied Mathematics ORIGINAL ARTICLE On the Double Moore–Gibson–Thompson System of Thermoviscoelasticity Filippo Dell’Oro1Lorenzo Liverani2Vittorino Pata1Ramon Quintanilla3 1Politecnico di Milano, Dipartimento di Matematica, Milano, Italy 2Department of Data Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany 3Departament de Matemàtiques, Universitat Politècnica de Catalunya, Terrassa, Barcelona, Spain Correspondence: Filippo Dell’Oro ([email protected]) Received: 13 May 2024 Revised: 10 October 2024 Accepted: 23 October 2024 Funding: L.L. has been supported by the Alexander von Humboldt Foundation. F.D. and V.P. have been partially supported by the Italian MIUR-PRIN Grant 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)”. Keywords: exponential stability | Moore–Gibson–Thompson system | solution semigroup | thermoviscoelasticity ABSTRACT In this paper, we address the system made by two coupled one-dimensional Moore–Gibson–Thompson equations { 𝑢+𝛼 𝑢−𝛽 𝑢𝑥𝑥 −𝛾𝑢𝑥𝑥 =𝑝(𝛼  𝑤𝑥+ 𝑤𝑥)  𝑤+ 𝛼 𝑤− 𝛽 𝑤𝑥𝑥 − 𝛾𝑤𝑥𝑥 = 𝑝( 𝛼 𝑢𝑥+ 𝑢𝑥) arising in the description of thermoviscoelastic materials. Here, 𝛼,𝛽,𝛾,  𝛼,  𝛽,  𝛾>0while 𝑝 𝑝>0. When both the MGT equations lie in the subcritical regime, that is, 𝛽−𝛾 𝛼>0and  𝛽− 𝛾  𝛼>0 we prove that the system generates an exponentially stable solution semigroup. This improves some recent results in the literature, where the exponential stability is attained only within either a stronger condition than subcriticality of both equations, or when 𝛼 and  𝛼are sufficiently close. The key idea is to deduce the exponential stability from that of a related system, made by two coupled equations of the viscoelasticity type. The latter result has also an independent interest. 2010 MATHEMATICS SUBJECT CLASSIFICATION: 35B40, 35Q74, 74D05, 74F05 1Preamble Thermoviscoelasticity deals with materials that, besides having a tendency to recover their original state (elastic effects), also exhibit mechanical viscous effects, as well as thermal dissipation. Different descriptions of viscoelasticity can be found in the literature, such as the Kelvin–Voigt or the Zener ones, the latter often referred to as viscoelasticity of Moore–Gibson–Thompson type, or models depending on the history of the deformation gradient. Likewise, several different types of thermal dissipation have been proposed through the years, mainly to overcome the paradox of the instantaneous propagation of thermal waves, typical of the classical Fourier law [1, 2]. With no claim to be exhaustive, we just recall the heat laws of Maxwell–Cattaneo [3, 4], Lord-Shulman [5], Gurtin–Pipkin [6], Green–Lindsay [7], Green–Naghdi [8–10], Moore–Gibson–Thompson [11–13]. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2024 The Author(s). Studies in Applied Mathematics published by Wiley Periodicals LLC. Studies in Applied Mathematics,2025;154:e12784 https://doi.org/10.1111/sapm.12784 1of10 In this paper, we focus on thermoviscoelastic materials where the dissipative effects, both viscous and thermal, are described by the Moore–Gibson–Thompson (MGT hereafter) equation which, writteninanabstractform,reads  𝑢+𝛼 𝑢+𝛽𝖠  𝑢+𝛾𝖠𝑢 =0, in the unknown 𝑢=𝑢(𝑡). Here, 𝖠is a strictly positive selfadjoint operator on some Hilbert space, while 𝛼,𝛽,𝛾 >0are given parameters, the dot standing for derivative with respect to time. Although originally introduced by Stokes in the mid-19th century [14], the MGT equation is named after the works [15, 16], and plays a paramount role in the description of several physical phenomena (see, e.g., [17–19], and references therein). For instance, from the mechanical point of view, it serves as a model of the Zener theory for viscous materials, whereas from the thermal point of view it can be obtained via the introduction of a relaxation time in the Green-Naghdi type III heat conduction theory. Another way to end up to with the MGT equation is to consider certain second-order equations with memory (e.g., the equation of viscoelasticity), for the particular choice of the negative exponential kernel [20]. This will be particularly relevant in connection with this paper. It is well known that the asymptotic behavior of the solutions to the MGT equation is strongly influenced by the values of the parameters 𝛼,𝛽,𝛾. Indeed, defining the stability number 𝜘=𝛽−𝛾 𝛼, the solutions decay to zero if and only if 𝜘>0, which defines the so-called subcritical regime of the model (see, e.g., [21, 22]). In fact, since the described phenomena (viscoelasticity and thermal conduction) are intrinsically dissipative, it will be of interest for our scopes to consider the case 𝜘>0only. 2The Double MGT System The main object of our investigation is the system made by two one-dimensional MGT equations coupled together, namely, { 𝑢+𝛼 𝑢−𝛽 𝑢𝑥𝑥 −𝛾𝑢𝑥𝑥 =𝑝(𝛼  𝑤𝑥+ 𝑤𝑥),  𝑤+ 𝛼 𝑤− 𝛽 𝑤𝑥𝑥 − 𝛾𝑤𝑥𝑥 = 𝑝( 𝛼 𝑢𝑥+ 𝑢𝑥), (2.1) in the unknowns 𝑢=𝑢(𝑥, 𝑡) and 𝑤=𝑤(𝑥,𝑡),with(𝑥, 𝑡) ∈ (0, 𝜋) ×ℝ+,where𝛼,𝛽,𝛾 and their corresponding hats are strictly positive constants, while 𝑝,  𝑝∈ℝ fulfill 𝑝 𝑝>0. Clearly, the interval (0, 𝜋) is taken just for convenience, and could be replaced by any other bounded open interval. System (2.1) is subject to the homogeneous Dirichlet boundary conditions 𝑢(0,𝑡) =𝑢(𝜋,𝑡) =𝑤(0,𝑡) =𝑤(𝜋,𝑡) =0, (2.2) and is supplemented with the initial conditions at the initial time 𝑡=0 ⎧ ⎪ ⎨ ⎪ ⎩ 𝑢(𝑥, 0) =𝑢0(𝑥),  𝑢(𝑥, 0) =𝑢1(𝑥),  𝑢(𝑥, 0) =𝑢2(𝑥), ⎧ ⎪ ⎨ ⎪ ⎩ 𝑤(𝑥,0) =𝑤0(𝑥),  𝑤(𝑥,0) =𝑤1(𝑥),  𝑤(𝑥,0) =𝑤2(𝑥), (2.3) where 𝑢𝚤and 𝑤𝚤are assigned functions on [0, 𝜋].Wealsorequire that the stability numbers of the two involved MGT equations be strictly positive, to wit, 𝜘=𝛽−𝛾 𝛼>0and 𝜘= 𝛽− 𝛾  𝛼>0. (2.4) This choice is consistent with the physical meaning of the two equations, which, as we will see shortly, are both dissipative. Remark 2.1. System (2.1)isessentially one-dimensional due to the particular nature of the coupling, which involves the first derivative in space. In fact, replacing 𝜕𝑥with (−Δ)1∕2, although in this case the physical meaning is less clear, the analysis carried out in this paper extends verbatim to the 𝑁-dimensional case. 3 Derivation of the Physical Model Given a thermoviscoelastic bar of linear mass density 𝜌>0 occupying the interval [0, 𝜋], the equations ruling the evolution of the transversal displacement 𝑢=𝑢(𝑥,𝑡) and the entropy 𝜂= 𝜂(𝑥,𝑡),with(𝑥, 𝑡) ∈ (0, 𝜋) ×ℝ,read {𝜌 𝑢=𝗍𝑥, 𝜏 𝜂=𝗊𝑥,(3.1) the subscript 𝑥standing for space derivative, where 𝗍=𝗍(𝑥, 𝑡) is the stress, 𝗊=𝗊(𝑥, 𝑡) is the heat flux, and 𝜏>0is the reference temperature which, without loss of generality, will be conventionally set equal to 1 hereafter. We will consider dissipative effects (viscous and thermal) which are due to the influence of the past history of the material on the future dynamics. From the classical axioms of thermomechanics, it is not actually immediate to propose a thermoviscoelastic theory complying with such an assumption. In this respect, one of the best and first appeared approaches is the linear theory of Gurtin [23], based on the invariance of entropy under time reversal. The distinctive character of this theory lies in the choice of the constitutive equations for 𝗍,𝜂,and𝗊, which, for the case of centrosymmetric materials, take the form (omitting the dependence on the space variable) 𝗍=∫𝑡 −∞ [𝑓(𝑡 −𝑠)  𝑢𝑥(𝑠) +𝑎(𝑡 −𝑠)  𝜃(𝑠)]𝑑𝑠, 𝜂=∫𝑡 −∞ [−𝑎(𝑡 −𝑠)  𝑢𝑥(𝑠) +𝑏(𝑡 −𝑠)  𝜃(𝑠)]𝑑𝑠, 𝗊=∫𝑡 −∞ 𝑔(𝑡 −𝑠)𝜃𝑥(𝑠)𝑑𝑠. Here, 𝜃=𝜃(𝑥,𝑡) is the relative temperature, while the convolution kernels 𝑓,𝑔,𝑎,𝑏, are positive convex decreasing functions, having strictly positive limit at infinity, which do not depend on the material point 𝑥. In fact, being a coupling term, the role of 𝑎and −𝑎can be interchanged. However, in this work we shall assume that the functions 𝑎and 𝑏are constant: namely, 𝑎≠0and 𝑏>0. Accordingly, within the physically reasonable hypotheses that both 𝑢and 𝜃vanish as 𝑡→−∞, plugging the constitutive 2of10 Studies in Applied Mathematics,2025 14679590, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/sapm.12784 by Readcube (Labtiva Inc.), Wiley Online Library on [04/12/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License equations into Equation (3.1) we deduce the system ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜌 𝑢−∫𝑡 −∞ 𝑓(𝑡 −𝑠)  𝑢𝑥𝑥(𝑠)𝑑𝑠 =𝑎𝜃𝑥, 𝑏 𝜃−∫𝑡 −∞ 𝑔(𝑡 −𝑠)𝜃𝑥𝑥(𝑠)𝑑𝑠 =𝑎 𝑢𝑥. (3.2) At this point, borrowing the concept from Green and Naghdi [8– 10], we introduce the thermal displacement 𝑤(𝑥,𝑡) =𝑤(𝑥,0) +∫𝑡 0 𝜃(𝑥,𝑠)ds, hence satisfying the relation  𝑤=𝜃. With this position, system (3.2) takes the more symmetric form ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜌 𝑢−∫𝑡 −∞ 𝑓(𝑡 −𝑠)  𝑢𝑥𝑥(𝑠)𝑑𝑠 =𝑎 𝑤𝑥, 𝑏 𝑤−∫𝑡 −∞ 𝑔(𝑡 −𝑠)  𝑤𝑥𝑥(𝑠)𝑑𝑠 =𝑎 𝑢𝑥. (3.3) The next step is to perform an integration by parts, together with a change of variables, yielding the equalities ∫𝑡 −∞ 𝑓(𝑡 −𝑠)  𝑢𝑥𝑥(𝑠)𝑑𝑠 =𝑓(∞)𝑢𝑥𝑥 −∫∞ 0 𝑓′(𝑠)[𝑢𝑥𝑥(𝑡) −𝑢𝑥𝑥(𝑡 −𝑠)]𝑑𝑠, ∫𝑡 −∞ 𝑔(𝑡 −𝑠)  𝑤𝑥𝑥(𝑠)𝑑𝑠 =𝑔(∞)𝑤𝑥𝑥 −∫∞ 0 𝑔′(𝑠)[𝑤𝑥𝑥(𝑡) −𝑤𝑥𝑥(𝑡 −𝑠)]𝑑𝑠, the prime denoting the derivative with respect to 𝑠. Then, defining the decreasing (and summable for physical reasons) nonnegative functions 𝜇(𝑠) =− 1 𝜌𝑓′(𝑠) and  𝜇(𝑠) =− 1 𝑏𝑔′(𝑠), and setting 𝜚=1 𝜌𝑓(∞),  𝜚=1 𝑏𝑔(∞), 𝑝 =𝑎 𝜌, 𝑝=𝑎 𝑏, we obtain from Equation (3.3) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩  𝑢−𝜚𝑢𝑥𝑥 −∫∞ 0 𝜇(𝑠)[𝑢𝑥𝑥(𝑡) −𝑢𝑥𝑥(𝑡 −𝑠)]𝑑𝑠 =𝑝 𝑤𝑥,  𝑤− 𝜚𝑤𝑥𝑥 −∫∞ 0  𝜇(𝑠)[𝑤𝑥𝑥(𝑡) −𝑤𝑥𝑥(𝑡 −𝑠)]𝑑𝑠 = 𝑝 𝑢𝑥. (3.4) System (3.4) is made by two equations of the viscoelasticity type (although here the second equation has a different physical meaning) coupled together, where all the physical parameters in play are free. Actually, the most significant convolution kernels 𝜇 and  𝜇from the physical viewpoint are the negative exponentials. Accordingly, let us take two kernels of the form 𝜇(𝑠) =𝜅 𝜀e−𝑠 𝜀and  𝜇(𝑠) = 𝜅  𝜀e−𝑠  𝜀,(3.5) for some 𝜅,  𝜅,𝜀,  𝜀>0. Then, taking the sum of the first equation of Equation (3.4)with𝜀-times its time derivative, and the sum of the second equation with  𝜀-times its time derivative, we end up with the system ⎧ ⎪ ⎨ ⎪ ⎩  𝑢+1 𝜀 𝑢−(𝜚 +𝜅)  𝑢𝑥𝑥 −𝜚 𝜀𝑢𝑥𝑥 =𝑝 𝜀 𝑤𝑥+𝑝 𝑤𝑥,  𝑤+1  𝜀 𝑤−( 𝜚+ 𝜅)  𝑤𝑥𝑥 − 𝜚  𝜀𝑤𝑥𝑥 = 𝑝  𝜀 𝑢𝑥+ 𝑝 𝑢𝑥, (3.6) where now all the memory terms have disappeared. Setting 𝛼=1 𝜀,𝛽=𝜚+𝜅, 𝛾 =𝜚 𝜀, 𝛼=1  𝜀, 𝛽= 𝜚+ 𝜅,  𝛾= 𝜚  𝜀, system (3.6) is nothing but Equation (2.1). Note that condition (2.4) is automatically satisfied, as the above choice yields 𝜘=𝛽−𝛾 𝛼=𝜅and 𝜘= 𝛽− 𝛾  𝛼= 𝜅. In conclusion, this procedure allows to recover all possible 𝛼,𝛽,𝛾,  𝛼,  𝛽,  𝛾>0only if both the MGT equations in Equation (2.1) lie in the subcritical regime, that is, only if Equation (2.4) holds. 4 Well-Posedness 4.1 Notation We denote by 𝐻the Hilbert space 𝐿2(0, 𝜋) with the inner product and norm ⟨⋅, ⋅⟩and ‖⋅‖, respectively. Calling by 𝖠=−𝜕𝑥𝑥 the strictly positive self-adjoint Laplace–Dirichlet operator on 𝐻, hence with dom(𝖠) =𝐻2(0,𝜋)∩𝐻 1 0(0, 𝜋), we will consider for 𝚤=−1, 1, 2 the further Hilbert spaces 𝐻𝚤=dom(𝖠 𝗂 𝟤), endowed with the inner product and norms ⟨𝑢,𝑣⟩𝚤=⟨𝖠𝗂 𝟤𝑢,𝖠 𝗂 𝟤𝑣⟩and ‖𝑢‖𝚤=‖𝖠𝗂 𝟤𝑢‖. Thus, 𝐻−1=𝐻−1(0, 𝜋), 𝐻1=𝐻1 0(0, 𝜋), 𝐻2=𝐻2(0, 𝜋) ∩ 𝐻1 0(0, 𝜋), the right-hand sides being the usual Sobolev spaces on (0, 𝜋).In particular, ‖𝑢𝑥‖−1=‖𝑢‖,‖𝑢‖1=‖𝑢𝑥‖,‖𝑢‖2=‖𝑢𝑥𝑥‖. We also recall the Poincaré inequality ‖𝑢‖≤‖𝑢‖1,∀𝑢∈𝐻 1. Here, the Poincaré constant equals 1 due to the choice of the interval (0, 𝜋). The dynamics of our system (2.1) will take place in the phase space =𝐻1×𝐻1×𝐻×𝐻1×𝐻1×𝐻, 3of10 14679590, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/sapm.12784 by Readcube (Labtiva Inc.), Wiley Online Library on [04/12/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License endowed with the standard product norm, denoted by ‖⋅‖.We will also encounter the more regular space 1=𝐻2×𝐻2×𝐻1×𝐻2×𝐻2×𝐻1. 4.2 The Solution Semigroup Introducing the six-component vector 𝒖=(𝑢, 𝑢∗,𝑢∗∗,𝑤,𝑤∗,𝑤∗∗), problem (2.1)–(2.2) can be written as the Ordinary Differential Equation (ODE) on  𝑑 𝑑𝑡𝒖(𝑡) =𝔸𝒖(𝑡), (4.1) while the initial condition (2.3) translates into 𝒖(0) =𝒖0=(𝑢0,𝑢 1,𝑢 2,𝑤 0,𝑤 1,𝑤 2)∈. The linear operator 𝔸is given by 𝔸(𝑢, 𝑢∗,𝑢∗∗,𝑤,𝑤∗,𝑤∗∗)=(𝑢∗,𝑢∗∗,𝑢∗∗∗,𝑤∗,𝑤∗∗,𝑤∗∗∗), where 𝑢∗∗∗ =−𝛼𝑢∗∗ +(𝛽𝑢∗+𝛾𝑢)𝑥𝑥 +𝑝(𝛼𝑤∗ 𝑥+𝑤∗∗ 𝑥), 𝑢∗∗∗ =−  𝛼𝑤∗∗ +( 𝛽𝑤∗+ 𝛾𝑤)𝑥𝑥 + 𝑝( 𝛼𝑢∗ 𝑥+𝑢∗∗ 𝑥), with domain dom(𝔸) =⎧ ⎪ ⎨ ⎪ ⎩ 𝒖∈|||||||| 𝑢∗∗,𝑤∗∗ ∈𝐻 1 𝛽𝑢∗+𝛾𝑢 ∈ 𝐻2  𝛽𝑤∗+ 𝛾𝑤 ∈ 𝐻2⎫ ⎪ ⎬ ⎪ ⎭ . Observe that we have the inclusion 1⊂dom(𝔸). Equation (2.1) turns out to generate a strongly continuous semigroup of bounded linear operators 𝑆(𝑡) ∶ →,whose infinitesimal generator is 𝔸. Accordingly, for all initial data 𝒖0∈ the solution at time 𝑡is given by 𝒖(𝑡) =𝑆(𝑡)𝒖0, whose related energy reads 𝖤(𝑡) =1 2‖𝑆(𝑡)𝒖0‖2 . The proof of this fact is detailed in [24] via semigroup techniques. However, it could also be obtained by means of energy estimates, performing the standard MGT multiplication [20], that is, multiplying in 𝐻the first equation in Equation (2.1)by  𝑢+𝛼 𝑢,andthe second one by  𝑤+ 𝛼 𝑤(times 𝑝∕  𝑝, in order to have a cancellation of the higher-order terms in the coupling). This, for regular initial data 𝒖0∈dom(𝔸),gives 𝑑 𝑑𝑡𝖥+𝛼𝜘‖ 𝑢‖2 1+𝑝  𝑝 𝛼𝜘‖ 𝑤‖2 1=𝖦, where 𝖥=1 2[𝛾 𝛼‖ 𝑢+𝛼𝑢‖2 1+‖ 𝑢+𝛼 𝑢‖2+𝜘‖ 𝑢‖2 1] +𝑝 2 𝑝[ 𝛾  𝛼‖ 𝑤+ 𝛼𝑤‖2 1+‖ 𝑤+ 𝛼 𝑤‖2+𝜘‖ 𝑤‖2 1] is equivalent to the standard energy functional 𝖤(see [20]for more details), while 𝖦=𝑝𝛼⟨ 𝑤𝑥, 𝑢+𝛼 𝑢⟩−𝑝𝛼⟨ 𝑢𝑥, 𝑤⟩+𝑝 𝛼⟨ 𝑢𝑥, 𝑤+ 𝛼 𝑤⟩−𝑝 𝛼⟨ 𝑤𝑥, 𝑢⟩. Note that 𝖦is controlled by 𝖥. This, by a straightforward application of the Gronwall lemma, yields 𝖤(𝑡) ≤𝐾1𝖤(0)e𝐾2𝑡,(4.2) for some positive constants 𝐾1,𝐾 2. However, at this stage, unless we are in the simpler situation 𝛼= 𝛼implying 𝖦=0(hence 𝐾2=0), we do not even know whether the energy (hence the semigroup) is bounded. Remark 4.1. By the same multiplication, this time in 𝐻1,itiseasy to see that 𝑆(𝑡) is a strongly continuous semigroup on the space 1as well. 5The Main Result The main result of the paper concerns with the exponential stability of 𝑆(𝑡). Theorem 5.1. Within assumption (2.4), there exist constants 𝑀≥1and 𝜔>0such that the exponential estimate ‖𝑆(𝑡)𝒖0‖≤𝑀‖𝒖0‖𝑒−𝜔𝑡 holds for every 𝑡≥0and every 𝒖0∈. Theorem 5.1 has been proved in [25] under the severe restriction that min{𝛼,  𝛼} >max {𝛾 𝛽, 𝛾  𝛽}. In other words, not only both the MGT equations must be subcritical, but they must remain subcritical even if the coefficients 𝛼and  𝛼are switched. Indeed, the key idea of [25] is to employ the same coefficient 𝛼(or  𝛼) for both equations, hence exploiting the same structure of the higher-order time derivatives, treating the remaining terms as a perturbation. Another important aspect is the role of the coupling and its connection with 𝛼and  𝛼.When 𝛼= 𝛼, the dissipative nature of the system emerges immediately from Equation (4.2), since in that case 𝐾2=0. On the contrary, when 𝛼and  𝛼are different, it is very difficult to exhibit the dissipative character of the model. In this situation, it has been shown in the very recent paper [24] that the solution semigroup 𝑆(𝑡) decays exponentially to zero whenever 𝛼and  𝛼are sufficiently close. Accordingly, the general problem for coefficients subject only to Equation (2.4) with no further restrictions requires a completely different approach, as the standard MGT estimates 4of10 Studies in Applied Mathematics,2025 14679590, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/sapm.12784 by Readcube (Labtiva Inc.), Wiley Online Library on [04/12/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License FIGURE 1 Scheme of the proof of Theorem 5.1. seem to unavoidably fail. The main idea of this work is to bypass completely this step, by exploiting in a deeper way the connection between our system (2.1)and(3.4), conventionally referred to as the double viscoelastic system hereafter. Indeed, in Sections 6 and 7, we will show that system (3.4) generates a strongly continuous semigroup 𝑇(𝑡) in a suitable functional framework, and such a semigroup is exponentially stable. Incidentally, this fact has an independent interest. Then, in Section 8, devoted to the proof of Theorem 5.1, we will show that part of the trajectories of the two semigroups coincide, and this will be enough to deduce the exponential stability of 𝑆(𝑡) from that of 𝑇(𝑡) (Figure 1). 6The Double Viscoelastic System We now turn our attention to system (3.4), where the convolution (or memory) kernels 𝜇and  𝜇areassumedtobenonnegative, nonincreasing, absolutely continuous, and summable function on ℝ+,oftotalmass ∫∞ 0 𝜇(𝑠)𝑑𝑠 =𝜅>0and ∫∞ 0  𝜇(𝑠)𝑑𝑠 = 𝜅>0. A particular example of kernels of this kind is given by Equation (3.5), but of course much more general kernels are possible, even exhibiting an integrable singularity about zero. The main feature of Equation (3.4) is that the convolution integrals act on the variables 𝑢and 𝑤for all times up to the actual time 𝑡. Therefore, being the initial time conventionally set at 𝑡=0,the functions 𝑢and 𝑤for negative times are assumed to be prescribed initialdata, whichneed not solvetheequation, butratherdescribe the past history of the system. As suggested by the seminal work [26](seealso[27, 28]), one way to handle Equation (3.4)is to introduce for 𝑡≥0and 𝑠>0the auxiliary variables 𝜂𝑡(𝑠) =𝑢(𝑡) −𝑢(𝑡 −𝑠) and 𝜉𝑡(𝑠) =𝑤(𝑡) −𝑤(𝑡 −𝑠), where we omitted the dependence on 𝑥. Accordingly, besides imposing the homogeneous Dirichlet boundary conditions on 𝑢,𝑤, 𝜂, 𝜉, we supplemented Equation (3.4) with the initial conditions ⎧ ⎪ ⎨ ⎪ ⎩ 𝑢(0) =𝑢0,  𝑢(0) =𝑢1, 𝜂0(𝑠) =𝜂0(𝑠), ⎧ ⎪ ⎨ ⎪ ⎩ 𝑤(0) =𝑤0,  𝑤(0) =𝑤1, 𝜉0(𝑠) =𝜉0(𝑠), (6.1) where 𝑢0,𝑢 1,𝜂 0,𝑤 0,𝑤 1,𝜉 0are assigned data. Thus, system (3.4) can be rewritten in the form ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩  𝑢−𝜚𝑢𝑥𝑥 −∫∞ 0 𝜇(𝑠)𝜂𝑥𝑥(𝑠)𝑑𝑠 =𝑝 𝑤𝑥,  𝑤− 𝜚𝑤𝑥𝑥 −∫∞ 0  𝜇(𝑠)𝜉𝑥𝑥(𝑠)𝑑𝑠 = 𝑝 𝑢𝑥, (6.2) where 𝜂𝑡(𝑠) ={𝑢(𝑡) −𝑢(𝑡 −𝑠) 0 <𝑠≤𝑡, 𝜂0(𝑠 −𝑡) +𝑢(𝑡) −𝑢0𝑠>𝑡, (6.3) and 𝜉𝑡(𝑠) ={𝑤(𝑡) −𝑤(𝑡 −𝑠) 0 <𝑠≤𝑡, 𝜉0(𝑠 −𝑡) +𝑤(𝑡) −𝑤0𝑠>𝑡. (6.4) In order to set this problem in the correct functional framework, we introduce the weighted 𝐿2-spaces of 𝐻1-valued functions =𝐿2 𝜇(ℝ+;𝐻1)and  =𝐿2  𝜇(ℝ+;𝐻1), normed by ‖𝜂‖=(∫∞ 0 𝜇(𝑠)‖𝜂(𝑠)‖2 1)1 2 and ‖𝜉‖ =(∫∞ 0  𝜇(𝑠)‖𝜉(𝑠)‖2 1)1 2 . Then, defining the phase space =𝐻1×𝐻××𝐻1×𝐻× , endowed with the norm ‖(𝑢, 𝑢∗,𝜂,𝑤,𝑤∗,𝜉)‖2 =𝜚‖𝑢‖2 1+‖𝑢∗‖2+‖𝜂‖2  +𝑝  𝑝[ 𝜚‖𝑤‖2 1+‖𝑤∗‖2+‖𝜉‖2  ], we view Equations (6.2)–(6.4) as an evolution system in . Actually, denoting by 𝖳the infinitesimal generator of the righttranslation semigroup on , that is, the linear operator [𝖳𝜂](𝑠) =−𝜂′(𝑠) with dom(𝖳) ={𝜂∈∶𝜂 ′∈,𝜂(0)=0}, and by  𝖳the analogous one on  , it is readily seen that Equations (6.3)and(6.4) are the mild solutions (in the sense of [29, 30]) to the evolution equations on and  , respectively,  𝜂=𝖳𝜂 + 𝑢and  𝜉= 𝖳𝜉 + 𝑤. 5of10 14679590, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/sapm.12784 by Readcube (Labtiva Inc.), Wiley Online Library on [04/12/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Accordingly, Equations (6.2)–(6.4) can be given the equivalent form ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩  𝑢−𝜚𝑢𝑥𝑥 −∫∞ 0 𝜇(𝑠)𝜂𝑥𝑥(𝑠)𝑑𝑠 =𝑝 𝑤𝑥,  𝜂=𝖳𝜂 + 𝑢,  𝑤− 𝜚𝑤𝑥𝑥 −∫∞ 0  𝜇(𝑠)𝜉𝑥𝑥(𝑠)𝑑𝑠 = 𝑝 𝑢𝑥,  𝜉= 𝖳𝜉 + 𝑤. (6.5) System (6.5) can be shown to generate a strongly continuous semigroup of linear contractions 𝑇(𝑡) ∶ →. Hence, for every initial datum 𝒗0=(𝑢0,𝑢 1,𝜂 0,𝑤 0,𝑤 1,𝜉 0)∈, its unique solution at time 𝑡is given by 𝒗(𝑡) =𝑇(𝑡)𝒗0, with corresponding energy 𝖤𝖵(𝑡) =1 2‖𝑇(𝑡)𝒗0‖2 .(6.6) The proof of this fact can be done along the same lines of [28], where the case of a single viscoelasticity equation is treated, actually for a much more general class of memory kernels, possibly exhibiting discontinuities. The key step is proving the energy equality, valid for all regular initial data (in particular, with 𝜂0and 𝜉0in the domains of 𝖳and  𝖳) 𝑑 𝑑𝑡𝖤𝖵−1 2∫∞ 0 𝜇′(𝑠)‖𝜂(𝑠)‖2 1𝑑𝑠 −𝑝 2 𝑝∫∞ 0  𝜇′(𝑠)‖𝜉(𝑠)‖2 1𝑑𝑠 =0. (6.7) Such an identity follows by a multiplication of Equation (6.5)by 𝒗(𝑡) in the phase space . 7 Exponential Decay of the Double Viscoelastic System The next step is showing that the semigroup 𝑇(𝑡) generated by system (6.5) is exponentially stable. To this end, we need some further assumptions on the memory kernels: as commonly done in the literature, we require that 𝜇′(𝑠) +𝛿𝜇(𝑠) ≤0and  𝜇′(𝑠) +𝛿 𝜇(𝑠) ≤0, for some 𝛿>0and almost every 𝑠>0. This yields the controls −∫∞ 0 𝜇′(𝑠)‖𝜂(𝑠)‖2 1𝑑𝑠 ≥𝛿‖𝜂‖2 and −∫∞ 0  𝜇′(𝑠)‖𝜉(𝑠)‖2 1𝑑𝑠 ≥𝛿‖𝜉‖2  .(7.1) The result reads as follows. Theorem 7.1. There exist constants 𝑄≥1and 𝜔>0such that the exponential estimate ‖𝑇(𝑡)𝒗0‖≤𝑄‖𝒗0‖𝑒−𝜔𝑡 holds for every 𝑡≥0and every 𝒗0∈. For simplicity, we shall assume hereafter that both 𝜇and  𝜇are bounded about zero, namely, 𝜇(0) =lim 𝑠→0 𝜇(𝑠) ∈ (0,∞)and  𝜇(0) =lim 𝑠→0  𝜇(𝑠) ∈ (0, ∞). The general case can be handled by introducing a suitable cutoff (see [28] for more details). For regular initial data 𝒗0= (𝑢0,𝑢 1,𝜂 0,𝑤 0,𝑤 1,𝜉 0),with𝜂0∈dom(𝖳)and 𝜉0∈dom( 𝖳),let 𝒗(𝑡) =𝑇(𝑡)𝒗0=(𝑢(𝑡),  𝑢(𝑡),𝜂𝑡, 𝑤(𝑡),  𝑤(𝑡),𝜉𝑡) be the corresponding (regular) solution to Equation (6.5). We introduce the auxiliary functional Φ(𝑡) =− 1 𝜅∫∞ 0 𝜇(𝑠)⟨𝜂𝑡(𝑠),  𝑢(𝑡)⟩𝑑𝑠 −𝑝  𝜅 𝑝∫∞ 0  𝜇(𝑠)⟨𝜉𝑡(𝑠),  𝑤(𝑡)⟩𝑑𝑠. By direct calculations, we find the identity 𝑑 𝑑𝑡 Φ+‖ 𝑢‖2+𝑝  𝑝‖ 𝑤‖2=−1 𝜅∫∞ 0 𝜇(𝑠)⟨𝖳𝜂(𝑠),  𝑢⟩𝑑𝑠 −𝑝  𝜅 𝑝∫∞ 0  𝜇(𝑠)⟨ 𝖳𝜉(𝑠),  𝑤⟩𝑑𝑠 (7.2) +1 𝜅∫∞ 0 𝜇(𝑠)(∫∞ 0 𝜇(𝜎)⟨𝜂(𝑠),𝜂(𝜎)⟩1𝑑𝜎)𝑑𝑠 +𝑝  𝜅 𝑝∫∞ 0  𝜇(𝑠)(∫∞ 0  𝜇(𝜎)⟨𝜉(𝑠),𝜉(𝜎)⟩1𝑑𝜎)𝑑𝑠 +𝜚 𝜅∫∞ 0 𝜇(𝑠)⟨𝜂(𝑠),𝑢⟩1𝑑𝑠 + 𝜚𝑝  𝜅 𝑝∫∞ 0  𝜇(𝑠)⟨𝜉(𝑠),𝑤⟩1𝑑𝑠 +𝑝 𝜅∫∞ 0 𝜇(𝑠)⟨𝜂𝑥(𝑠),  𝑤⟩𝑑𝑠 +𝑝  𝜅∫∞ 0  𝜇(𝑠)⟨𝜉𝑥(𝑠),  𝑢⟩𝑑𝑠. Integrating by parts in 𝑠(the boundary terms vanish, see [28]), we infer that −1 𝜅∫∞ 0 𝜇(𝑠)⟨𝖳𝜂(𝑠),  𝑢⟩𝑑𝑠 =− 1 𝜅∫∞ 0 𝜇′(𝑠)⟨𝜂(𝑠),  𝑢⟩𝑑𝑠 ≤1 4‖ 𝑢‖2+1 𝜅2(∫∞ 0 −𝜇′(𝑠)‖𝜂(𝑠)‖𝑑𝑠)2 ≤1 4‖ 𝑢‖2−𝜇(0) 𝜅2∫∞ 0 𝜇′(𝑠)‖𝜂(𝑠)‖2 1𝑑𝑠, where the last bound follows from the Hölder and the Poincaré inequalities, which will repeatedly used hereafter, often without mention. Similarly, −𝑝  𝜅 𝑝∫∞ 0  𝜇(𝑠)⟨ 𝖳𝜉(𝑠),  𝑤⟩𝑑𝑠 ≤𝑝 4 𝑝‖ 𝑤‖2− 𝜇(0)𝑝  𝜅2 𝑝∫∞ 0  𝜇′(𝑠)‖𝜉(𝑠)‖2 1𝑑𝑠. Moreover 1 𝜅∫∞ 0 𝜇(𝑠)(∫∞ 0 𝜇(𝜎)⟨𝜂(𝑠),𝜂(𝜎)⟩1𝑑𝜎)𝑑𝑠 ≤1 𝜅(∫∞ 0 𝜇(𝑠)‖𝜂(𝑠)‖1𝑑𝑠)2≤‖𝜂‖2 , and, by the same token, 𝑝  𝜅 𝑝∫∞ 0  𝜇(𝑠)(∫∞ 0  𝜇(𝜎)⟨𝜉(𝑠),𝜉(𝜎)⟩1𝑑𝜎)𝑑𝑠 ≤𝑝  𝑝‖𝜉‖2  . 6of10 Studies in Applied Mathematics,2025 14679590, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/sapm.12784 by Readcube (Labtiva Inc.), Wiley Online Library on [04/12/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Next, it is readily seen that 𝜚 𝜅∫∞ 0 𝜇(𝑠)⟨𝜂(𝑠),𝑢⟩1𝑑𝑠 + 𝜚𝑝  𝜅 𝑝∫∞ 0  𝜇(𝑠)⟨𝜉(𝑠),𝑤⟩1𝑑𝑠 ≤𝐶‖𝑢‖1‖𝜂‖+𝐶‖𝑤‖1‖𝜉‖ . Here and in the sequel of this proof, 𝐶>0stands for a generic structural constant, independent of the initial data, which may vary even within the same line. Finally, 𝑝 𝜅∫∞ 0 𝜇(𝑠)⟨𝜂𝑥(𝑠),  𝑤⟩𝑑𝑠 +𝑝  𝜅∫∞ 0  𝜇(𝑠)⟨𝜉𝑥(𝑠),  𝑢⟩𝑑𝑠 ≤1 4‖ 𝑢‖2+𝑝 4 𝑝‖ 𝑤‖2+𝐶‖𝜂‖2 +𝐶‖𝜉‖2  . Substituting all the estimates obtained so far into Equation (7.2), we arrive at 𝑑 𝑑𝑡Φ+1 2‖ 𝑢‖2+𝑝 2 𝑝‖ 𝑤‖2≤𝐶‖𝑢‖1‖𝜂‖+𝐶‖𝑤‖1‖𝜉‖  +𝐶‖𝜂‖2 +𝐶‖𝜉‖2   −𝐶∫∞ 0 𝜇′(𝑠)‖𝜂(𝑠)‖2 1𝑑𝑠 −𝐶∫∞ 0  𝜇′(𝑠)‖𝜉(𝑠)‖2 1𝑑𝑠. (7.3) At this point, we consider the further functional Ψ(𝑡) =⟨ 𝑢(𝑡),𝑢(𝑡)⟩+𝑝  𝑝⟨ 𝑤(𝑡),𝑤(𝑡)⟩. A straightforward computation entails the equality 𝑑 𝑑𝑡 Ψ+𝜚‖𝑢‖2 1+ 𝜚𝑝  𝑝‖𝑤‖2 1=− ∫∞ 0 𝜇(𝑠)⟨𝜂(𝑠),𝑢⟩1𝑑𝑠 −𝑝  𝑝∫∞ 0  𝜇(𝑠)⟨𝜉(𝑠),𝑤⟩1𝑑𝑠 +‖ 𝑢‖2+𝑝  𝑝‖ 𝑤‖2−𝑝⟨ 𝑤,𝑢𝑥⟩−𝑝⟨ 𝑢,𝑤𝑥⟩. It is not difficult to check that the right-hand side above is controlled by 𝐶‖𝑢‖1‖𝜂‖+𝐶‖𝑤‖1‖𝜉‖ +𝐶‖ 𝑢‖2+𝐶‖ 𝑤‖2+𝐶‖ 𝑤‖‖𝑢‖1+𝐶‖ 𝑢‖‖𝑤‖1 ≤𝜚 2‖𝑢‖2 1+ 𝜚𝑝 2 𝑝‖𝑤‖2 1+𝐶‖ 𝑢‖2+𝐶‖ 𝑤‖2+𝐶‖𝜂‖2 +𝐶‖𝜉‖2  . Therefore, we end up with 𝑑 𝑑𝑡Ψ+𝜚 2‖𝑢‖2 1+ 𝜚𝑝 2 𝑝‖𝑤‖2 1≤𝐶‖ 𝑢‖2+𝐶‖ 𝑤‖2+𝐶‖𝜂‖2 +𝐶‖𝜉‖2  . (7.4) In order to finish the proof, we introduce for all 𝜀∈(0,1)the perturbed energy functional Λ𝜀(𝑡) =𝖤𝖵(𝑡) +𝜀3 2Φ(𝑡) +𝜀2Ψ(𝑡), where 𝖤𝖵has been defined in Equation (6.6). Since |Φ|≤𝐶𝖤𝖵and |Ψ|≤𝐶𝖤𝖵,uptotaking𝜀small enough we have 1 2𝖤𝖵≤Λ𝜀≤2𝖤𝖵.(7.5) In addition, collecting Equations (7.3)and(7.4) and the energy identity (6.7), we obtain 𝑑 𝑑𝑡Λ𝜀−(1 2−𝐶𝜀3 2)∫∞ 0 𝜇′(𝑠)‖𝜂(𝑠)‖2 1𝑑𝑠 −(𝑝 2 𝑝−𝐶𝜀3 2)∫∞ 0  𝜇′(𝑠)‖𝜉(𝑠)‖2 1𝑑𝑠 +1 2[𝜀2𝜚‖𝑢‖2 1+𝜀3 2‖ 𝑢‖2]+𝑝 2 𝑝[𝜀2 𝜚‖𝑤‖2 1+𝜀3 2‖ 𝑤‖2] ≤𝐶𝜀3 2‖𝑢‖1‖𝜂‖+𝐶𝜀3 2‖𝑤‖1‖𝜉‖ +𝐶𝜀2‖ 𝑢‖2+𝐶𝜀2‖ 𝑤‖2 +𝐶𝜀3 2‖𝜂‖2 +𝐶𝜀3 2‖𝜉‖2  . For all 𝜀sufficiently small, an exploitation of Equation (7.1) yields −(1 2−𝐶𝜀 3 2)∫∞ 0 𝜇′(𝑠)‖𝜂(𝑠)‖2 1𝑑𝑠 −(𝑝 2 𝑝−𝐶𝜀 3 2)∫∞ 0  𝜇′(𝑠)‖𝜉(𝑠)‖2 1𝑑𝑠 ≥𝛿 4[‖𝜂‖2 +𝑝  𝑝‖𝜉‖2  ]. Moreover, 𝐶𝜀3 2‖𝑢‖1‖𝜂‖+𝐶𝜀3 2‖𝑤‖1‖𝜉‖ +𝐶𝜀2‖ 𝑢‖2+𝐶𝜀2‖ 𝑤‖2 +𝐶𝜀3 2‖𝜂‖2 +𝐶𝜀3 2‖𝜉‖2   ≤𝐶𝜀5 2‖𝑢‖2 1+𝐶𝜀2‖ 𝑢‖2+𝐶𝜀5 2‖𝑤‖2 1+𝐶𝜀2‖ 𝑤‖2 +𝐶𝜀1 2‖𝜂‖2 +𝐶𝜀1 2‖𝜉‖2   ≤1 4[𝜀2𝜚‖𝑢‖2 1+𝜀3 2‖ 𝑢‖2]+𝑝 4 𝑝[𝜀2 𝜚‖𝑤‖2 1+𝜀3 2‖ 𝑤‖2] +𝛿 8[‖𝜂‖2 +𝑝  𝑝‖𝜉‖2  ]. In conclusion, fixing 𝜀>0small enough, there exists a structural constant 𝜔>0such that 𝑑 𝑑𝑡Λ𝜀+4𝜔𝖤𝖵≤0. Invoking Equation (7.5) and the Gronwall lemma, from the inequality above we infer that 𝖤𝖵(𝑡) ≤4𝖤𝖵(0)e−2𝜔𝑡, for every 𝑡≥0. By density, the conclusion holds for every initial data 𝒗0∈. Remark 7.2. Theorem 7.1 is actually true for a more general class of memory kernels, including kernels exhibiting (even infinitely many) discontinuity points. This requires minor changes in the proof, borrowing the techniques from [28]. 7of10 14679590, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/sapm.12784 by Readcube (Labtiva Inc.), Wiley Online Library on [04/12/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 8Proof of Theorem 5.1 Let us start from system (2.1) within assumption (2.4). Throughout this section, we will consider Equation (6.5) for the particular choice of the parameters and the memory kernels 𝜚=𝛾 𝛼, 𝜚= 𝛾  𝛼,𝜇(𝑠)=𝛼𝜘e−𝛼𝑠, 𝜇(𝑠) = 𝛼𝜘e− 𝛼𝑠. We begin to establish a link between the two systems. Lemma 8.1. Let 𝒖=(𝑢,  𝑢,  𝑢,𝑤,  𝑤,  𝑤) be the solution to Equation (2.1) corresponding to the initial datum 𝒖0= (𝑢0,𝑢 1,𝑢 2,𝑤 0,𝑤 1,𝑤 2)∈, and let 𝒗0=(𝑢0,𝑢 1,𝜂 0,𝑤 0,𝑤 1,𝜉 0)∈ , having set 𝜂0=1 𝜘[−𝖠−1𝑢2−𝛾 𝛼𝑢0+𝑝𝖠−1𝑤1𝑥],(8.1) 𝜉0=1 𝜘[−𝖠−1𝑤2− 𝛾  𝛼𝑤0+ 𝑝𝖠−1𝑢1𝑥].(8.2) Then 𝒗=(𝑢,  𝑢,𝜂, 𝑤,  𝑤,𝜉), with 𝜂and 𝜉given by Equations (6.3) and (6.4), respectively, is the solution to Equation (6.5) corresponding to the initial datum 𝒗0. Proof. Let us denote by 𝒘=(𝑟,  𝑟,𝜙,𝑦,  𝑦,𝜓) the unique solution to Equation (6.5) with initial datum 𝒗0. We observe that  𝑟(0) =𝑢2and  𝑦(0) =𝑤2. Indeed, since 𝒘solves Equation (6.5), its third component 𝜙is given by 𝜙𝑡(𝑠) ={𝑟(𝑡) −𝑟(𝑡 −𝑠) 0 <𝑠≤𝑡, 𝜂0+𝑟(𝑡) −𝑢0𝑠>𝑡, as 𝜂0(𝑠) is constant in 𝑠. Hence, the first equation of Equation (6.5), written in the variable 𝑟,𝑦 in place of 𝑢, 𝑤,reads  𝑟(𝑡) −𝛽𝑟𝑥𝑥(𝑡) +𝛼𝜘∫𝑡 0 e−𝛼𝑠𝑟𝑥𝑥(𝑡 −𝑠)𝑑𝑠 =𝑝 𝑦𝑥(𝑡) +𝜘(𝜂0𝑥𝑥 −𝑢0𝑥𝑥)e−𝛼𝑡. (8.3) For 𝑡=0, we obtain  𝑟(0) =𝛽𝑢0𝑥𝑥 +𝑝𝑤1𝑥 +𝜘(𝜂0𝑥𝑥 −𝑢0𝑥𝑥)=𝑢2, due to Equation (8.1). The equality  𝑦(0) =𝑤2is derived in the same way by using Equation (8.2). Next, we show that (𝑟,  𝑟,  𝑟,𝑦,  𝑦,  𝑦) solves Equation (2.1). To this end, we multiply Equation (8.3)bye 𝛼𝑡,toget [ 𝑟(𝑡) −𝛽𝑟𝑥𝑥(𝑡)]e𝛼𝑡 +𝛼𝜘∫𝑡 0 e𝛼𝑠𝑟𝑥𝑥(𝑠)𝑑𝑠 =𝑝 𝑦𝑥(𝑡)e𝛼𝑡 +𝜘(𝜂0𝑥𝑥 −𝑢0𝑥𝑥). Taking the time derivative, and dividing the resulting equation by e𝛼𝑡, we arrive at  𝑟+𝛼 𝑟−𝛽 𝑟𝑥𝑥 −𝛾𝑟𝑥𝑥 =𝑝(𝛼  𝑦𝑥+ 𝑦𝑥). Similarly, we obtain the corresponding equation for 𝑦.By uniqueness, we have the desired equality 𝒘=𝒗.□ In what follows, 𝑀≥1will denote a generic structural constant, independent of the initial data, which may vary even within the same line. Lemma 8.2. For every 𝒖0and 𝒗0as in the statement of Lemma 8.1, ‖𝒗0‖2 ≤𝑀[‖𝑢0‖2 1+‖𝑢1‖2+‖𝑢2‖2 −1+‖𝑤0‖2 1+‖𝑤1‖2 +‖𝑤2‖2 −1]≤𝑀‖𝒖0‖2 . Proof. The latter is an immediate consequence of the Poincaré inequality. As for the first one, it clearly suffices to show that ‖𝜂0‖2 +‖𝜉0‖2  ≤𝑀[‖𝑢0‖2 1+‖𝑢1‖2+‖𝑢2‖2 −1+‖𝑤0‖2 1 +‖𝑤1‖2+‖𝑤2‖2 −1]. Let us focus on 𝜂0. On account of Equation (8.1), noting again that 𝜂0does not depend on 𝑠, ‖𝜂0‖2 =1 𝜘‖‖‖−𝖠−1𝑢2−𝛾 𝛼𝑢0+𝑝𝖠−1𝑤1𝑥‖‖‖ 2 1≤𝑀[‖𝖠−1𝑢2‖2 1 +‖𝑢0‖2 1+‖𝖠−1𝑤1𝑥‖2 1]. But ‖𝖠−1𝑢2‖1=‖𝑢2‖−1and ‖𝖠−1𝑤1𝑥‖1=‖𝑤1‖. Repeating the argument for 𝜉0, the claim is established. □ We have now all the ingredients to complete the proof of Theorem 5.1. Let us choose an arbitrary, but more regular, initial datum for Equation (2.1), that is, 𝒖0=(𝑢0,𝑢 1,𝑢 2,𝑤 0,𝑤 1,𝑤 2)∈1. Then, consider system (6.5) with initial datum 𝒗0as in Lemma 8.1. Exploiting the exponential stability of 𝑇(𝑡), ensured by Theorem 7.1, we draw the estimate ‖𝑢(𝑡)‖2 1+‖𝑤(𝑡)‖2 1≤𝑀‖𝑇(𝑡)𝒗0‖2 ≤𝑀‖𝒗0‖2 e−2𝜔𝑡, and by Lemma 8.2 we infer that ‖𝑢(𝑡)‖2 1+‖𝑤(𝑡)‖2 1≤𝑀‖𝒖0‖2 e−2𝜔𝑡.(8.4) The next step is to take the time derivative of Equation (2.1), observing that the functions 𝑣= 𝑢and 𝑧= 𝑤 continue to solve Equation (2.1), but this time for the initial datum 𝒚0=(𝑢1,𝑢 2,𝑢 3,𝑤 1,𝑤 2,𝑤 3), where 𝑢3=−𝛼𝑢2+𝛽𝑢1𝑥𝑥 +𝛾𝑢0𝑥𝑥 +𝑝(𝛼𝑤1𝑥 +𝑤2𝑥), 𝑤3=−  𝛼𝑤2+ 𝛽𝑤1𝑥𝑥 + 𝛾𝑤0𝑥𝑥 + 𝑝( 𝛼𝑢1𝑥 +𝑢2𝑥). 8of10 Studies in Applied Mathematics,2025 14679590, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/sapm.12784 by Readcube (Labtiva Inc.), Wiley Online Library on [04/12/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Notice that 𝒚0∈, due to the assumed higher regularity of 𝒖0. Besides, in light of the Poincaré inequality, ‖𝑢3‖2 −1+‖𝑤3‖2 −1≤𝑀‖𝒖0‖2 .(8.5) At this point, we apply once again Lemma 8.1 with 𝒚0in place of 𝒖0. Accordingly, in place of 𝒗0we will have 𝒛0=(𝑢1,𝑢 2,𝜂 1,𝑤 1,𝑤 2,𝜉 1), where 𝜂1=1 𝜘[−𝖠−1𝑢3−𝛾 𝛼𝑢1+𝑝𝖠−1𝑤2𝑥], 𝜉1=1 𝜘[−𝖠−1𝑤3− 𝛾  𝛼𝑤1+ 𝑝𝖠−1𝑢2𝑥]. Due to the first inequality of Lemma 8.2 together with Equation (8.5), ‖𝒛0‖2 ≤𝑀[‖𝑢1‖2 1+‖𝑢2‖2+‖𝑢3‖2 −1+‖𝑤1‖2 1+‖𝑤2‖2 +‖𝑤3‖2 −1]≤𝑀‖𝒖0‖2 . Therefore, since  𝑣= 𝑢and  𝑧= 𝑤, by a further application of Theorem 7.1 we conclude that ‖ 𝑢(𝑡)‖2 1+‖ 𝑢(𝑡)‖2+‖ 𝑤(𝑡)‖2 1+‖ 𝑤(𝑡)‖2≤𝑀‖𝑇(𝑡)𝒛0‖2 ≤𝑀‖𝒖0‖2 e−2𝜔𝑡. Collecting this inequality and Equation (8.4), we end up with the final estimate ‖𝑆(𝑡)𝒖0‖≤𝑀‖𝒖0‖e−𝜔𝑡, valid for every 𝒖0∈1. By density, such an estimate remains valid for every 𝒖0∈. This finishes the proof. Acknowledgments L.L. has been supported by the Alexander von Humboldt Foundation. 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