Abstract
The Rao–Nakra sandwich beam models under study in this paper consist of coupled two wave equations and one Euler–Bernoulli equation on an open bounded interval of \(\mathbb {R}\) subject to homogeneous Dirichlet–Neumann boundary conditions. The objective of this paper is to investigate the well-posedness and asymptotic behavior of solutions when a single internal infinite memory term is present either (i) on one wave equation or (ii) on the Euler–Bernoulli equation. The involved operator in the memory term is \(\left( -\partial _{xx}\right) ^m\) with \(m\in \{0,1\}\) in case (i), and \(m\in \{0,1,2\}\) in case (ii). It is proved in this paper that the system is well posed and can be indirectly stabilized polynomially (for strong solutions) independently from the parameter m, and the coefficients of the system in case (i), and if and only if the speeds of wave propagation of the two wave equations are different in case (ii), where the considered memory kernel is exponentially decreasing. We prove also that the same conditions guarantee the strong stability of the system (for weak solutions). However, in both cases (i) and (ii) and independently from the parameter m and the coefficients of the system, we prove that this single internal infinite memory term does not stabilize exponentially the whole considered Rao–Nakra sandwich beam models. On the other hand, the polynomial decay rates are given explicitly in terms of the parameter m. Our results are new because they are the first to demonstrate exponential, polynomial, and strong stability for Rao–Nakra sandwich beams with a single internal infinite memory.
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References
Akil M, Liu Z (2023) Stabilization of the generalized Rao–Nakra beam by partial viscous damping. Math Methods Appl Sci 46:1479–1510
Allen A, Hansen S (2009) Analyticity of a multilayer Mead–Markus plate. Nonlinear Anal Theory Methods Appl 71:1835–1842
Allen A, Hansen S (2010) Analyticity and optimal damping for a multilayer Mead–Markus sandwich beam. Discrete Continuous Dyn Syst 14:1279–1292
Ammari K, Komornik V, Sepúlveda M, Vera O (2024) Stability of the Rao–Nakra sandwich beam with a dissipation of fractional derivative type: theoretical and numerical study. arXiv:2405.18619
Arendt W, Batty CJK (1988) Tauberian theorems and stability of one one-parameter semigroups. Trans Am Math Soc 306:837–852
Arendt W, Batty CJK, Hieber M, Neubrander F (2011) Vector-valued Laplace transforms and Cauchy problems. Birkhäuser, Basel
Astudillo M, Oquendo HP (2021) Stability Results for a Timoshenko system with a fractional operator in the memory. Appl Math Optim 83:1247–1275
Batty CJK, Duyckaerts T (2008) Non-uniform stability for bounded semi-groups on Banach spaces. J Evol Equ 8:765–780
Batty CJK, Chill R, Tomilov Y (2016) Fine scales of decay of operator semigroups. J Eur Math Soc 18:853–929
Bekhouche R, Guesmia A, Messaoudi S (2022) Uniform and weak stability of Bresse system with one infinite memory in the shear angle displacements. Arab J Math 11:155–178
Borichev A, Tomilov Y (2010) Optimal polynomial decay of functions and operator semigroups. Math Ann 347:455–478
Brezis H (2011) Functional analysis. Universitex, Sobolev spaces and partial differential equations. Springer, Berlin
Cabanillas V, Raposo C, Potenciano-Machado L (2022) Stability of solution for Rao–Nakra sandwich beam model with Kelvin-Voigt damping and time delay. Theoret Appl Mech 49:71–84
Dafermos CM (1970) Asymptotic stability in viscoelasticity. Arch Rat Mech Anal 37:297–308
Dell Oro F (2021) On the stability of Bresse and Timoshenko systems with hyperbolic heat conduction. J Diff Equ 281:148–198
Dell Oro F, Laeng E, Pata V (2017) A quantitative Riemann–Lebesgue lemma with application to equations with memory. Proc Ame Math Soc 145:2909–2915
Feng B, Ma TF, Monteiro RN, Raposo CA (2018) Dynamics of laminated Timoshenko beam. J Dyn Diff Equ 30:1489–1507
Feng B, Raposo CA, Nonato CA, Soufyane A (2023) Analysis of exponential stabilization for Rao–Nakra sandwich beam with time-varying weight and time-varying delay: multiplier method versus observability. Math Control Relat Fields 13:631–663
Gilbarg D, Trudinger NS (2001) Elliptic partial differential equations of second order. Springer-Verlag, Berlin
Giorgi C, Vegni FM (2004) Uniform energy estimates for a semilinear evolution equation of the Mindlin–Timoshenko beam with memory. Math Comput Model 39:1005–1021
Guesmia A (2017) Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement. Nonauton Dyn Syst 4:78–97
Guesmia A (2020) On the stability of a laminated Timoshenko problem with interfacial slip in the whole space under frictional dampings or infinite memories. Nonauton Dyn Syst 7:194–218
Guesmia A (2020) Well-posedness and stability results for laminated Timoshenko beams with interfacial slip and infinite memory. IMA J Math Control Inform 37:300–350
Guesmia A, Kafini M (2015) Bresse system with infinite memories. Math Methods Appl Sci 38:2389–2402
Guesmia A, Kirane M (2016) Uniform and weak stability of Bresse system with two infinite memories. Z Angew Math Phys 67:1–39
Guesmia A, Messaoudi SA (2014) A general stability result in a Timoshenko system with infinite memory: a new approach. Math Methods Appl Sci 37:384–392
Guesmia A, Messaoudi SA, Soufyane A (2012) Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-heat systems. Electr J Diff Equ 2012:1–45
Guesmia A, Mohamad-Ali Z, Wehbe A, Youssef W (2023) Polynomial stability of a transmission problem involving Timoshenko systems with fractional Kelvin–Voigt damping. Math Methods Appl Sci 46:7140–7176
Hansen SW, Rajaram R (2005) Riesz basis property and related results for a Rao–Nakra sandwich beam. Discr Continuous Dyn Syst 365–375
Hansen SW, Rajaram R (2005) Simultaneous boundary control of a Rao-Nakra sandwich beam. In: Proceedings of 44th IEEE conference on decision and control and European control conference, 3146–3151
Hansen SW, Imanuvilov OY (2011) Exact controllability of a multilayer Rao–Nakra plate with free boundary conditions. Math Control Relat Fields 1:189–230
Hansen SW, Imanuvilov OY (2011) Exact controllability of a multilayer Rao–Nakra Plate with clamped boundary conditions. ESAIM Control Optim Calc Var 17:1101–1132
Hansen S, Liu Z (1999) Analyticity of semigroup associated with a laminated composite beam. Springer, Boston, pp 47–54
Hansen SW, Spies R (1997) Structural damping in a laminated beam due to interfacial slip. J Sound Vib 204:183–202
Huang FL (1985) Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces. Ann Diff Equ 1:43–56
Li Y, Liu Z, Whang Y (2018) Weak stability of a laminated beam. Math Control Relat Fields 8:789–808
Liu ZY, Rao BP (2005) Characterization of polynomial decay rate for the solution of linear evolution equation. Z Angew Math Phys 56:630–644
Liu Z, Trogdon SA, Yong J (1999) Modeling and analysis of a laminated beam. Math Comput Model 30:149–167
Liu Z, Rao B, Zhang Q (2020) Polynomial stability of the Rao–Nakra beam with a single internal viscous damping. J Diff Equ 269:6125–6162
Liu Z, Zheng S (1999) Semigroups associated with dissipative systems, 398 Research Notes in Mathematics. Chapman & Hall CRC, London
Mead D, Markus S (1969) The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions. J Sound Vib 10:163–175
Mukiawa SE (2023) Well-posedness and stabilization of a type three layer beam system with Gurtin–Pipkin’s thermal law. AIMS Math 8:28188–28209
Mukiawa SE, Enyi CD, Audu JD (2022) Well-posedness and stability result for a thermoelastic Rao–Nakra beam model. J Thermal Stresses 45:720–739
Muñoz Rivera JE, Sare HDF (2008) Stability of Timoshenko systems with past history. J Math Anal Appl 339:482–502
Özkan Özer A, Hansen SW (2013) Uniform stabilization of a multilayer Rao–Nakra sandwich beam. Evol Equ Control Theory 2:695–710
Özkan Özer A, Hansen SW (2014) Exact boundary controllability results for a multilayer Rao–Nakra sandwich beam. SIAM J Control Optim 52:1314–1337
Pata V, Zucchi A (2001) Attractors for a damped hyperbolic equation with linear memory. Adv Math Sci Appl 11:505–529
Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York
Prüss J (1984) On the spectrum of \(C_0\) semigroups. Trans Am Math Soc 284:847–857
Rajaram R (2007) Exact boundary controllability result for a Rao–Nakra sandwich beam. Syst Control Lett 56:558–567
Rao YS, Nakra BC (1974) Vibrations of unsymmetrical sanwich beams and plates with viscoelastic cores. J Sound Vibr 34:309–326
Raposo CA (2021) Rao–Nakra model with internal damping and time delay. Math Moravica 25:53–67
Raposo CA, Vera Villagran OP, Ferreira J, Piskin E (2021) Rao–Nakra sandwich beam with second sound. Part Diff Equ Appl Math 4:1–5
Rozendaal J, Seifert D, Stahn R (2019) Optimal rates of decay for operators semigroups on Hilbert spaces. Adv Math 346:359–388
Vera O, Raposo CA, Nonato AC, Ramos AJ (2022) Stability of solution for Rao–Nakra Sandwich beam with boundary dissipation of fractional derivative type. J Fract Calc Appl 13:116–143
Wang Y (2019) Boundary feedback stabilization of a Rao-Nakra sandwich beam. J Phys Conf Ser 1324
Yan MJ, Dowell EH (1972) Governing equations for vibrating constrained-layer damping sandwich plates and beams. J Appl Mech 39:1041–1046
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The author would like to express his gratitude to the anonymous referee for very careful reading and punctual suggestions.
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Guesmia, A. Study of the well-posedness and decay rates for Rao–Nakra sandwich beam models subject to a single internal infinite memory and Dirichlet–Neumann boundary conditions. Comp. Appl. Math. 44, 67 (2025). https://doi.org/10.1007/s40314-024-03033-6
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DOI: https://doi.org/10.1007/s40314-024-03033-6
Keywords
- Rao–Nakra sandwich beam
- Infinite memory
- Well-posedness
- Asymptotic behavior
- Semigroups theory
- Energy method
- Frequency domain approach
- Spectral analysis