On a coupled system of viscoelastic wave equation of infinite memory with acoustic boundary conditions
DOI:
https://doi.org/10.24193/subbmath.2024.1.11Keywords:
Viscoelastic damping, acoustic boundary conditions, well-posedness, exponential stabilityAbstract
This work deals with a coupled system of viscoelastic wave equation of infinite memory with mixed Dirichlet-Neumann boundary conditions. The coupling is via the acoustic boundary conditions on a portion of the boundary. The semigroup theory is used to show the well-posedness and regularity of the initial and boundary value problem. Moreover, we investigate exponential stability of the system taking into account Gearhart-Prüss’s theorem.
Mathematics Subject Classification (2010): 35A01, 74B05, 93D15.
Received 11 June 2021; Accepted 13 October 2021
References
Alabau-Boussouira, F., Stabilisation frontière indirecte de systèmes faiblement couplés, C.R. Acad. Sci. Paris Sér. I, 328(1999), no. 11, 1015-1020.
Alabau-Boussouira, F., Cannarsa, P., Komornik, V., Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2(2002), no. 2, 127-150.
Almeida, R.G.C., Santos, M.L., Lack of exponential decay of a coupled system of wave equations with memory, Nonlinear Anal. RWA, 12(2011), no. 2, 1023-1032.
Apalara, T.A., Raposo, C.A., Nonato, C.A.S., Exponential stability for laminated beams with a frictional damping, Arch. Math., 114(2020), no. 4, 471-480.
Beale, J.T., Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25(1976), no. 9, 895-917.
Benomar, K., Benaissa, A., Optimal decay rates for the acoustic wave motions with boundary memory damping, Stud. Univ. Babeș-Bolyai Math., 65(2020), no. 3, 471-482.
Borichev, A., Tomilov, Y., Optimal polynomial decay of functions and operator semi- groups, Math. Ann., 347(2009), no. 2, 455-478.
Boukhatem, Y., Benabderrahmane, B., Existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions, Nonlinear Anal. TMA, 97(2014), 191- 209.
Cordeiro, S.M.S., Lobato, R.F.C., Raposo, C.A., Optimal polynomial decay for a coupled system of wave with past history, Open J. Math. Anal., 4(2020), no. 1, 49-59.
Frota, C.L., Larkin, N.A., Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains, in Contributions to nonlinear analysis, (T. Cazenave et al., Ed.), Progr. Nonlinear Differential Equations Appl. 66, Birkhäuser, Basel, 2005, 297-312.
Gao, Y., Liang, J., Xiao, T. J., A new method to obtain uniform decay rates for multidimensional wave equations with nonlinear acoustic boundary conditions, SIAM J. Control Optim. 56(2018), no. 2, 1303–1320.
Komornik, V., Bopeng, R., Boundary stabilization of compactly coupled wave equations, Asymptot. Anal., 14(1997), no. 4, 339-359.
Limam, A., Boukhatem, Y., Benabderrahmane, B., New general stability for a variable coefficients thermo-viscoelastic coupled system of second sound with acoustic boundary conditions, Comput. Appl. Math., 40(2021), no. 3, 88.
Liu, Z., Zheng, S., Semigroups Associated with Dissipative Systems, CRC Press, 1999.
Morse, P.M., Ingard, K.U., Theoretical Acoustics, Princeton University Press, Princeton, NJ, 1986.
Peralta, G.R., Stabilization of the wave equation with acoustic and delay boundary conditions, Semigroup Forum, 96(2018), no. 2, 357-376.
Prüss, J., On the spectrum of C0−semigroups, Tran. Amer. Math. Soc., 284(1984), no. 2, 847-850.
Prüss, J., Decay properties for the solutions of a partial differential equation with memory, Arch. Math., 92(2009), no. 2, 158-173.
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