Logarithmic decay rate for a structural-acoustic interaction model with supercritical source and nonlinear damping

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Abstract

In this paper, we study a structural acoustics model with supercritical source and nonlinear damping. The model consists of a wave equation defined on a bounded domain which is strongly coupled with a Berger plate equation acting on the elastic wall. The aim of the paper is to remove the strong condition $u \in L^{\infty}\left(\mathbb{R}^{+} ; L^{\frac{3}{2}}(\Omega)\right) \text { for } m_{1}>5$ in the work [1] to obtain a weaker energy decay, and give a logarithmic decay rate under a weak assumption on the initial data.

Keywords

structural acoustics model / energy decay / supercritical sources / nonlinear damping

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References

[1] B. Feng, Y. Guo, M. A. Rammaha, On the asymptotic behavior of solutions to a structural acoustics model, J. Differ. Equations, 372 (2023), 315-347. https://doi.org/10.1016/j.jde.2023.06.047

[2] H. T. Banks, W. Fang, R. J. Sicox, R. C. Smith, Approximation Methods for Control of Acoustics/Structure Models with Piezoceramic Actuators, Contract Report 189578, NASA, 1991.

[3] J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917. https://www.jstor.org/stable/24891055

[4] J. T. Beale, S. I. Rosencrans, Acoustic boundary conditions, Bull. Am. Math. Soc., 80 (1974), 1276-1278. https://doi.org/10.1090/S0002-9904-1974-13714-6

[5] M. S. Howe, Acoustics of Fluid-Structure Interactions, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1998. https://doi.org/10.1017/CBO9780511662898

[6] P. M. Morse, K. U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968.

[7] I. Lasiecka, Mathematical Control Theory of Coupled PDEs, Society for Industrial and Applied Mathematics (SIAM), 2002. https://doi.org/10.1137/1.9780898717099

[8] G. Avalos, I. Lasiecka, Uniform decay rates for solutions to a structural acoustics model with nonlinear dissipation, Appl. Math. Comput. Sci., 8 (1998), 287-312.

[9] G. Avalos, I. Lasiecka, Exact controllability of structural acoustic interactions, J. Math. Pures Appl., 82 (2003), 1047-1073. https://doi.org/10.1016/S0021-7824(03)00016-3

[10] G. Avalos, I. Lasiecka, Exact controllability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls, Adv. Differ. Equations, 10 (2005), 901-930. https://doi.org/10.57262/ade/1355867823

[11] J. Cagnol, I. Lasiecka, C. Lebiedzik, J. P. Zolesio, Uniform stability in structural acoustic models with flexible curved walls, J. Differ. Equations, 186 (2002), 88-121. https://doi.org/10.1016/S0022-0396(02)00029-3

[12] I. Lasiecka, Boundary stabilization of a 3-dimensional structural acoustic model, J. Math. Pures Appl., 78 (1999), 203-232. https://doi.org/10.1016/S0021-7824(01)80009-X

[13] V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differ. Equations, 109 (1994), 295-308. https://doi.org/10.1006/jdeq.1994.1051

[14] E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differ. Equations, 186 (2002), 259-298. https://doi.org/10.1016/S0022-0396(02)00023-2

[15] M. M. Cavalcanti, V. N. D. Cavalcanti, I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differ. Equations, 236 (2007), 407-459. https://doi.org/10.1016/j.jde.2007.02.004

[16] L. Bociu, I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differ. Equations, 249 (2010), 654-683. https://doi.org/10.1016/j.jde.2010.03.009

[17] L. Bociu, I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860. https://doi.org/10.3934/dcds.2008.22.835

[18] L. Bociu, I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Appl. Math., 35 (2008), 281-304. https://doi.org/10.4064/am35-3-3

[19] Y. Guo, M. A. Rammaha, Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc., 366 (2014), 2265-2325. https://www.jstor.org/stable/23813106

[20] Y. Guo, M. A. Rammaha, Blow-up of solutions to systems of nonlinear wave equations with supercritical sources, Appl. Anal., 92 (2013), 1101-1115. https://doi.org/10.1080/00036811.2011.649734

[21] Y. Guo, M. A. Rammaha, Global existence and decay of energy to systems of wave equations with damping and supercritical sources, Z. Angew. Math. Phys., 64 (2013), 621-658. https://doi.org/10.1007/s00033-012-0252-6

[22] A. R. Becklin, M. A. Rammaha, Global solutions to a structure acoustic interaction model with nonlinear sources, J. Math. Anal. Appl., 487 (2020), 123977. https://doi.org/10.1016/j.jmaa.2020.123977

[23] V. N. D. Cavalcanti, M. M. Cavalcanti, T. D. Marchiori, C. M. Webler, Asymptotic behaviour of the energy to the viscoelastic wave equation with localized hereditary memory and supercritical source term, J. Dyn. Differ. Equations, 35 (2023), 3381-3431. https://doi.org/10.1007/s10884-022-10142-5

[24] Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. S. Titi, D. Toundykov, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differ. Equations, 257 (2014), 3778-3812. https://doi.org/10.1016/j.jde.2014.07.009

[25] Y. Guo, M. A. Rammaha, S. Sakuntasathien, Energy decay of a viscoelastic wave equation with supercritical nonlinearities, Z. Angew. Math. Phys., 69 (2018), 65. https://doi.org/10.1007/s00033-018-0961-6

[26] Y. Guo, M. A. Rammaha, S. Sakuntasathien, Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities, J. Differ. Equations, 262 (2017), 1956-1979. https://doi.org/10.1016/j.jde.2016.10.037

[27] M. Liao, Energy decay of wave equations with infinite memory effects versus supercritical frictional dampings, Ric. Mat., 74 (2025), 1037-1050. https://doi.org/10.1007/s11587-023-00832-7

[28] M. Liao, B. Feng, A piezoelectric beam model with nonlinear interior dampings and supercritical sources, Discrete Contin. Dyna. Syst., 52 (2026), 293-341. http://doi.org/10.3934/dcds.2026053

[29] P. Pei, M. A. Rammaha, D. Toundykov, Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources, J. Math. Phys., 56 (2015), 081503. https://doi.org/10.1063/1.4927688

[30] M. A. Rammaha, Z. Wilstein, Hadamard well-posedness for wave equations with p-Laplacian damping and supercritical sources, Adv. Differ. Equations, 17 (2012), 105-150. https://doi.org/10.57262/ade/1355703099

[31] A. R. Becklin, M. A. Rammaha, Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms, Evol. Equations Control Theory, 10 (2021), 797-836. https://doi.org/10.3934/eect.2020093

[32] B. Feng, Y. Guo, M. A. Rammaha, Blow-up theorems for a structural acoustics model, J. Math. Anal. Appl., 529 (2024), 127600. https://doi.org/10.1016/j.jmaa.2023.127600

[33] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM: Control Optim. Calc. Var., 4 (1999), 419-444. https://doi.org/10.1051/cocv:1999116

[34] A. Haraux, L. Tebou, Energy decay estimates for the wave equation with supercritical nonlinear damping, J. Math. Anal. Appl., 550 (2025), 129622. https://doi.org/10.1016/j.jmaa.2025.129622