Abstract
We consider a wave equation with nonlinear acoustic boundary conditions. This is a nonlinearly coupled system of hyperbolic equations modeling an acoustic/structure interaction, with an additional boundary damping term to induce both existence of solutions as well as stability. Using the methods of Lasiecka and Tataru for a wave equation with nonlinear boundary damping, we demonstrate well-posedness and uniform decay rates for solutions in the finite energy space, with the results depending on the relationship between (i) the mass of the structure, (ii) the nonlinear coupling term, and (iii) the size of the nonlinear damping. We also show that solutions (in the linear case) depend continuously on the mass of the structure as it tends to zero, which provides rigorous justification for studying the case where the mass is equal to zero.
Similar content being viewed by others
References
Alber H., Cooper J.: Quasilinear hyperbolic 2 x 2 systems with a free, damping boundary condition. Journal für die reine und angewandte Mathematik 406, 10–43 (1990)
Avalos G.: The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics. Abstract and Applied Analysis 1, 203–217 (1996)
Avalos G., Lasiecka I.: The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system. Semigroup Forum 57, 278–292 (1998)
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, 1976.
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)
Beale J. T., Rosencrans S. I.: Acoustic boundary conditions. Bull. Amer. Math. Soc. 80, 1276–1278 (1974)
Cousin A. T., Frota C. L., Larkin N. A.: Global solvability and asymptotic behaviour of hyperbolic problem with acoustic boundary conditions. Funkcial. Ekvac. 44, 471–485 (2001)
Cousin A. T., Frota C. L., Larkin N. A.: On a system of klein-gordon type equations with acoustic boundary conditions. Journal of Mathematical Analysis and Applications 293, 293–309 (2004)
Frota C. L., Larkin N. A.: Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains. Progress in Nonlinear Differential Equations and Their Applications 66, 297–312 (2005)
Gal C., Goldstein G., Goldstein J.: Oscillatory boundary conditions for acoustic wave equations. Journal of Evolution Equations 3, 623–635 (2003)
Graber P.: Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system. Nonlinear Analysis: Theory and Applications 73, 3058–3068 (2010)
Graber P.: Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions. Nonlinear Analysis: Theory and Applications 74, 3058–3068 (2011)
H. Koch and I. Lasiecka, Hadamard wellposedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, vol. 50 of Progress in Nonlinear Differential Equations, 2002, pp. 197–217.
Lasiecka I.: Mathematical Control Theory of Coupled PDEs. SIAM, Philadelphia (2002)
Lasiecka I., Tataru D.: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential and Integral Equations 6, 507–533 (1993)
Lasiecka I., Triggiani R.: Uniform stabilization of the wave equation with dirichlet or neumann feedback control without geometrical conditions. Applied Mathematics and Optimization 25, 189–224 (1992)
Morse P. M., Ingard K. U.: Theoretical Acoustics. McGraw-Hill, New York (1968)
J. Y. Park and T. G. Ha, Well-posedness for the klein gordon equation with damping term and acoustic boundary conditions, J. Math. Physics, 50 (2009).
Sell G., You Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)
R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, vol. 49 of Mathematical Surveys and Monographs, AMS, 1997.
Temam R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin-Heidelberg-New York (1988)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by the Jefferson Scholars Foundation and the Virginia Space Grant Consortium.
Rights and permissions
About this article
Cite this article
Graber, P.J. Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping. J. Evol. Equ. 12, 141–164 (2012). https://doi.org/10.1007/s00028-011-0127-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-011-0127-x