Abstract
For wave equations with variable coefficients on regions which are not necessarily smooth, we study the energy decay rate when a nonlinear damping is applied on a suitable subrigion.
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Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, controll and stabilization of waves from the boundary. SIAM J. Cont. Optim., 30, 1024–1065 (1992)
Komornik, V.: Exact Controllability and Stabilization, Research in Applied Mathematics (Series Editors: P. G. Ciarlet and J. Lions), Masson/John Wiley Copublication, 1994
Komornik, V.: On the nonlinear boundary stabilization of the wave equation. Chin. Ann. of Math, 14B(2), 153–164 (1993)
Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary condition. Diff. Int. Eqs., 6, 507–533 (1993)
Lions, J. L.: Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev., 30, 1–68 (1988)
Wyler, A.: Stability of wave equations with dissipative boundary conditions in a bounded domain. Differential & and Integral Equations, 7(2), 345–366 (1994)
Zuazua, E.: Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Cont. Optim., 28, 466–477 (1990)
Yao, P. F.: On the observability inequality for exact controllability of wave equations with variable coefficients. SIAM J. Cont. Optim., 37, 1568–1599 (1999)
Lagnese, J.: Control of wave processes with distributed controls supported on a subregion. SIAM J. Cont. Optim., 21, 68–85 (1983)
Liu, K.: Locally distributed control and damping for the conservative systems. SIAM J. Cont. Optim., 35, 1574–1590 (1997)
Tcheugoué Tébou, L. R.: Stabilization of the wave equation with localized nonlinear damping. J. Differential Equations, 145, 502–524 (1998)
Wu, H., Shen, L. L., Yu, Y. L.: Introduction to Riemannian Geometry (in chinese), Beijing University Press, Beijing 1989
Feng, S., Feng, D.: Locally distributed control of wave equation with variable coefficients. Science in China, Series F, 44, 309–315 (2001)
Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces, Nord-hoff, International Publishing, Bucuresti, România, 1976
Aubin, J. P.: Un Théorème de compacité. C. R. Acad. Sci., 56, 5042–5044 (1963)
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Supported by the NSFC (Grant No. 6174008)
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Feng, S.J., Feng, D.X. Nonlinear Internal Damping of Wave Equations with Variable Coefficients. Acta Math Sinica 20, 1057–1072 (2004). https://doi.org/10.1007/s10114-004-0394-3
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DOI: https://doi.org/10.1007/s10114-004-0394-3