# zbMATH — the first resource for mathematics

Attractors for semilinear damped wave equations with an acoustic boundary condition. (English) Zbl 1239.35025
Summary: We study a semilinear weakly damped wave equation equipped with an acoustic boundary condition. The problem can be considered as a system consisting of the wave equation describing the evolution of an unknown function $$u=u(x,t)$$, $$x \in\Omega$$, in the domain coupled with an ordinary differential equation for an unknown function $$\delta=\delta (x,t)$$, $$x \in \Gamma:= \partial\Omega$$, on the boundary. A compatibility condition is also added due to physical reasons. This problem is inspired by a model originally proposed by J. T. Beale and S. I. Rosencrans [Bull. Am. Math. Soc. 80, 1276–1278 (1974; Zbl 0294.35045)]. The goal of the paper is to analyze the global asymptotic behavior of the solutions. We prove the existence of an absorbing set and of the global attractor in the energy phase space. Furthermore, the regularity properties of the global attractor are investigated. This is a difficult issue since standard techniques based on the use of fractional operators cannot be exploited. Finally, we prove the existence of an exponential attractor. The analysis is carried out in dependence of two damping coefficients.

##### MSC:
 35B41 Attractors 35B40 Asymptotic behavior of solutions to PDEs 35L71 Second-order semilinear hyperbolic equations 37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents 35L20 Initial-boundary value problems for second-order hyperbolic equations
Full Text:
##### References:
 [1] Ball J.M.: Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10, 31–52 (2004) · Zbl 1056.37084 · doi:10.3934/dcds.2004.10.31 [2] Ball J.M.: Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Amer. Math. Soc. 64, 370–373 (1977) · Zbl 0353.47017 [3] Beale J.T., Rosencrans S.I.: Acoustic boundary conditions. Bull. Amer. Math. Soc. 80, 1276–1278 (1974) · Zbl 0294.35045 · doi:10.1090/S0002-9904-1974-13714-6 [4] Beale J.T.: Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25, 895–917 (1976) · Zbl 0332.35050 · doi:10.1512/iumj.1976.25.25071 [5] Beale J.T.: Acoustic scattering from locally reacting surfaces. Indiana Univ. Math. J. 26, 199–222 (1977) · Zbl 0352.35071 · doi:10.1512/iumj.1977.26.26015 [6] Belleri V., Pata V.: Attractors for semilinear strongly damped wave equation on $${$$\backslash$$mathbb{R}\^n}$$ . Discrete Contin. Dyn. Syst. 7, 719–735 (2001) · Zbl 1200.35032 · doi:10.3934/dcds.2001.7.719 [7] Casarino V., Engel K.J., Nagel R., Nickel G.: A semigroup approach to boundary feedback systems. Integral Equations Operator Theory 47, 289–306 (2003) · Zbl 1048.47054 · doi:10.1007/s00020-002-1163-2 [8] Casarino V., Engel K.J., Nickel G., Piazzera S.: Decoupling techniques for wave equations with dynamic boundary conditions. Discrete Contin. Dyn. Syst. Ser. B 12, 761–772 (2005) · Zbl 1082.34048 · doi:10.3934/dcds.2005.12.761 [9] Conti M., Pata V.: Weakly dissipative semilinear equations of viscoelasticity. Commun. Pure Appl. Anal. 4, 705–720 (2005) · Zbl 1101.35016 · doi:10.3934/cpaa.2005.4.705 [10] Efendiev M., Miranville A., Zelik S.: Exponential attractors for a nonlinear reaction-diffusion system in $${$$\backslash$$mathbb{R}\^3}$$ . C. R. Acad. Sci. Paris Sr. I Math. 330, 713–718 (2000) · Zbl 1151.35315 · doi:10.1016/S0764-4442(00)00259-7 [11] Fabrie P., Galusinski C., Miranville A., Zelik S.: Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete Contin. Dyn. Syst. 10, 211–238 (2004) · Zbl 1060.35011 · doi:10.3934/dcds.2004.10.211 [12] Frigeri S.: Asymptotic behavior of a hyperbolic system arising in ferroelectricity. Commun. Pure Appl. Anal. 7, 1393–1414 (2008) · Zbl 1152.35319 · doi:10.3934/cpaa.2008.7.1393 [13] S. Frigeri, Long time behavior of some semilinear hyperbolic systems, Ph. D. Thesis, in preparation. · Zbl 1152.35319 [14] Frota C.L., Goldstein J.A.: Some nonlinear wave equations with acoustic boundary conditions. J. Differential Equations 164, 92–109 (2000) · Zbl 0979.35105 · doi:10.1006/jdeq.1999.3743 [15] Gal C.G., Goldstein G.R., Goldstein J.A.: Oscillatory boundary conditions for acoustic wave equations. J. Evol. Equ. 3, 623–636 (2003) · Zbl 1058.35139 · doi:10.1007/s00028-003-0113-z [16] Gatti S., Grasselli M., Miranville A., Pata V.: On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation. J. Math. Anal. Appl. 312, 230–247 (2006) · Zbl 1160.35518 · doi:10.1016/j.jmaa.2005.03.029 [17] Ghidaglia J.M.: A note on the strong convergence towards attractors for damped forced KdV equations. J. Differential Equations 110, 356–359 (1994) · Zbl 0805.35114 · doi:10.1006/jdeq.1994.1071 [18] Goubet O.: Regularity of the attractor for the weakly damped nonlinear Schrödinger equation. Appl. Anal. 60, 99–119 (1996) · Zbl 0872.35100 · doi:10.1080/00036819608840420 [19] Goubet O., Moise I.: Attractor for dissipative Zakharov system. Nonlinear Anal. 31, 823–847 (1998) · Zbl 0902.35090 · doi:10.1016/S0362-546X(97)00441-0 [20] Grasselli M., Miranville A., Pata V., Zelik S.: Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials. Math. Nachr. 280, 1475–1509 (2007) · Zbl 1133.35017 · doi:10.1002/mana.200510560 [21] Grasselli M., Pata V.: Asymptotic behavior of a parabolic-hyperbolic system. Commun. Pure Appl. Anal. 3, 849–881 (2004) · Zbl 1079.35022 · doi:10.3934/cpaa.2004.3.849 [22] Lions J.-L., Magenes E.: Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, New-York (1972) · Zbl 0223.35039 [23] Moise I., Rosa R.: On the regularity of the global attractor of a weakly damped forced Korteweg-de Vries equation. Adv. Differential Equation 2, 257–296 (1997) · Zbl 1023.35525 [24] Moise I., Rosa R., Wang X.: Attractors for non-compact semigroups via energy equation. Nonlinearity 11, 1369–1393 (1998) · Zbl 0914.35023 · doi:10.1088/0951-7715/11/5/012 [25] Morse P.M., Ingard K.U.: Theoretical acoustics. McGraw-Hill, New York (1968) [26] Mugnolo D.: Abstract wave equations with acoustic boundary conditions. Math. Nachr. 279, 299–318 (2006) · Zbl 1109.47035 · doi:10.1002/mana.200310362 [27] Pata V., Squassina M.: On the strongly damped wave equation. Comm. Math. Phys. 253, 511–533 (2005) · Zbl 1068.35077 · doi:10.1007/s00220-004-1233-1 [28] Pata V., Zelik S.: A remark on the damped wave equation. Commun. Pure Appl. Anal. 5, 609–614 (2006) · Zbl 1140.35533 [29] Rosa R.: The global attractor for the 2d Navier-Stokes flow on some unbounded domains. Nonlinear Anal. 32, 71–85 (1998) · Zbl 0901.35070 · doi:10.1016/S0362-546X(97)00453-7 [30] Wang X.: An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications. Phys. D 88, 167–175 (1995) · Zbl 0900.35372 · doi:10.1016/0167-2789(95)00196-B [31] Yeoul P.J., Ae K.J.: Some nonlinear wave equations with nonlinear memory source term and acoustic boundary conditions. Numer. Funct. Anal. Optim. 27, 889–903 (2006) · Zbl 1106.76062 · doi:10.1080/01630560600884596 [32] Zelik S.: Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete Contin. Dyn. Syst. 11, 351–392 (2004) · Zbl 1059.35018 · doi:10.3934/dcds.2004.11.351 [33] S. Zheng, Nonlinear Evolution Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, 133, CHAPMAN and HALL/CRC, Boca Raton, Florida, 2004.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.