Abstract
This paper is concerned with the long time dynamics of the semilinear wave equation including localized interior damping term under acoustic boundary conditions. With appropriate assumptions, the existence of a global attractor for the semigroup generated by the considered problem is proved. Moreover, imposing more strict conditions on the nonlinearity in the equation, the regularity and finite dimensionality of the attractor are established.
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Morse, P.M., Ingard, K.U.: Theoretical Acoustics. McGraw Hill, New York (1968)
Beale, J.T., Rosencrans, S.T.: Acoustic boundary conditions. Bull. Am. Math. Soc. 80, 1276–1278 (1974)
Beale, J.T.: Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25, 895–917 (1976)
Beale, J.T.: Acoustic scattering from locally reacting surfaces. Indiana Univ. Math. J. 26, 199–222 (1977)
Frota, C.L., Goldstein, J.A.: Some nonlinear wave equations with acoustic boundary conditions. J. Differ. Equ. 164, 92–109 (2000)
Goldstein, G.R., Gal, C.G., Goldstein, J.A.: Oscillatory boundary conditions for acoustic wave equations. J. Evol. Equ. 3, 623–636 (2003)
Mugnolo, D.: Abstract wave equations with acoustic boundary conditions. Math. Nachr. 279, 299–318 (2006)
Vicente, A.: Wave equations with acoustic/memory boundary conditions. Bol. Soc. Paran. Mat. 27(1), 29–39 (2009)
Frota, C.L., Limaco, J., Clark, H., Medeiros, L.A.: On an evolution equation with acoustic boundary conditions. Math. Methods Appl. Sci. 34(16), 2047–2059 (2011)
Avalos, G., Lasiecka, I.: Uniform decay rates of solutions to a structural acoustic model with nonlinear dissipation. App. Math. Comp. Sci. 8, 287–312 (1998)
Munoz Rivera, J.E., Qin, Y.: Polynomial decay for the energy with an acoustic boundary condition. Appl. Math. Lett. 16, 249–256 (2003)
Avalos, G., Lasiecka, I.: Exact controllability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls. Adv. Differ. Equ. 10(8), 901–930 (2005)
Frota, C.L., Vicente, A.: A hyperbolic system of Klein Gordon type with acoustic boundary conditions. Int. J. Pure Appl. Math. 47, 185–198 (2008)
Gerbi, S., Said-Houari, B.: Local existence and exponential growth for a semilinear damped wave equation with dynamical boundary conditions. Adv. Differ. Equ. 13, 1051–1074 (2008)
Park, J.Y., Park, S.H.: Decay rate estimates for wave equations of memory type with acoustic boundary conditions. Nonlinear Anal. 74, 993–998 (2011)
Said-Houari, B., Graber, P.J.: On the wave equation with semilinear porous acoustic boundary conditions. J. Differ. Equ. 252, 4898–4941 (2012)
Vicente, A., Frota, C.L.: Uniform stabilization of wave equation with localized damping and acoustic boundary condition. J. Math. Anal. Appl. 436, 639–660 (2016)
Lee, M.J., Park, J.Y., Kang, Y.H.: Energy decay rates for the semilinear wave equation with memory boundary condition and acoustic boundary conditions. Comput. Math. Appl. 73, 1975–1986 (2017)
Frota, C.L., Vicente, A.: Uniform stabilization of wave equation with localized internal damping and acoustic boundary condition. Z. Angew. Math. Phys. 69, 85 (2018)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Frota, C.L., Vicente, A.: Stability for semilinear wave equation in an inhomogeneous medium with frictional localized damping and acoustic boundary conditions. SIAM J. Control. Optim. 58(4), 2411–2445 (2020)
Frigeri, S.: Attractors for semilinear damped wave equations with an acoustic boundary condition. J. Evol. Equ. 10, 29–58 (2010)
Frigeri, S.: On the Convergence to Stationary Solutions for a Semilinear Wave Equation with an Acoustic Boundary Condition. Z. Anal. Anwend. 30(2), 181–191 (2011)
Schomberg, J.L.: Attractors for damped semilinear wave equations with singularly perturbed acoustic boundary conditions. Elec. J. Differ. Eq. 152, 1–33 (2018)
Schomberg, J.L.: Attractors for damped semilinear wave equations with a Robin-Acoustic boundary perturbation. Nonlinear Anal. 189, 111582 (2019)
Ball, J.M.: Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Am. Math. Soc. 64, 370–373 (1977)
Pata, V., Zelik, S.: A remark on the damped wave equation. Commun. Pure Appl. Anal. 5, 609–614 (2006)
Ruiz, A.: Unique continuation for weak solutions of the wave equation plus a potential. J. Math Pures Appl. 710, 455–467 (1992)
Khanmamedov, A.: Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain. J. Differ. Equ. 225, 528–548 (2006)
Chueshov, I., Lasiecka, I.: Von Karman Evolution Equations. Springer, Berlin (2010)
Khanmamedov, AKh.: Global attractors for 2-D wave equations with displacement dependent damping. Math. Methods Appl. Sci. 33, 177–187 (2010)
Lions, J.L.: Controlabilité exacte, perturbations et stabilisation de systèmes distribués, tome 1. Masson, Paris (1988)
Zuazua, E.: Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures Appl. 70, 513–529 (1991)
Babin, A.V., Vishik, M.I.: Attractors for Evolution Equations. North-Holland, Amsterdam (1992)
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Sen, Z., Yayla, S. Long time behavior of semilinear wave equation with localized interior damping term under acoustic boundary condition. Japan J. Indust. Appl. Math. 40, 1221–1257 (2023). https://doi.org/10.1007/s13160-023-00571-0
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DOI: https://doi.org/10.1007/s13160-023-00571-0