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Proyecciones (Antofagasta)
versão impressa ISSN 0716-0917
Proyecciones (Antofagasta) v.23 n.3 Antofagasta dez. 2004
doi: 10.4067/S0716-09172004000300002
| Proyecciones UNIFORM STABILIZATION OF A PLATE EQUATION WITH NONLINEAR LOCALIZED DISSIPATION
ADEMIR F. PAZOTO LUCICLÉIA COELHO RUY COIMBRA CHARAO ABSTRACT We study the existence and uniqueness of a plate equation in a bounded domain of Rn, with a dissipative nonlinear term, localized in a neighborhood of part of the boundary of the domain. We use techniques from control theory, the unique continuation property and Nakao method to prove the uniform stabilization of the energy of the system with algebraic decay rates depending on the order of the nonlinearity of the dissipative term. References [1] Alabau, F., Komornik. V., Boundary observability, controlability and stabilization of linear elastodynamics systems, SIAM J. Control Optim. 37 (1999), 521-542. [ Links ] [2] E. Bisognin, V. Bisognin, R. Coimbra Charao, Uniform stabilization for elastic waves system with highly nonlinear localized dissipation. Portugaliae Matemática 60, 99-124, 2003. [ Links ] [3] H. Brezis, Análisis funcional Teoria y aplicaciones. Alianza Editorial, Madrid, 1983. [ Links ] [4] Coddington, E., Levinson, N., Theory of ordinary differential equations, McGraw-Hill, New York, 1955. [ Links ] [5] Guesmia, A., On the decay estimates for elasticity systems with some localized dissipations, Asymptotic Analysis 22 (2000), 1-13. [ Links ] [6] A. Haraux, Semigroupes Linéaires et équations dévolution linéaires périodiques. Université Pierre et Marie Curie, Paris, 1978. [ Links ] [7] Horn, M. A., Nonlinear boundary stabilization of a system of anisotropic elasticity with light internal damping, Contemporary Mathematics 268 (2000), 177-189. Uniform Stabilization of a Plate Equation 233 [ Links ] [8] Komornik, V., Exact controllability and stabilization, the multiplier method, John Wiley Sons - Masson, Paris, 1994. [ Links ] [9] J. U. Kim, A unique continuation property of a beam equation with variable coe.cients, in estimation and control of distributed parameter sustems. (Desch, W., Kappel, F., Kunisch, K. eds.), 197-205, Birkhäuser, Basel, 1991 (Internationel Series of numerical mathematics, 100 (1991)). [ Links ] [10] J. L. Lions, Exact Controllability, Stabilization and perturbations for Distributed Systems. SIAM Rev.30, 1-68, 1988. [ Links ] [11] J. L. Lions, Quelques M´ethodes de R´esolution des Probl´emes aux Limites Non Linéaires. Gauthier - Villars, Paris, 1969. [ Links ] [12] Martinez, P., Decay of solutions of the wave equation with a local highly degenerate dissipation, Asymptotic Analysis 19 (1999), 1-17. [ Links ] [13] Nakao, M., Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann. 305 (1996), 403-417. [ Links ] [14] M. Tucsnak, Stabilization of Bernoulli - Euler beam by means of a pointwise feedback force. Siam J. control Optim 39, n4, 1160-1181, 2000. [ Links ] [15] T´ebou, L. R. T., Well-posedness and energy decay estimates for the damped wave equation with Lr localizing coe.cient, Comm. in Partial Di.. Eqs. 23 (1998), 1839-1855. [ Links ] [16] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Comm.PartialDiff. Eqs. 15, 205-235, 1990. [ Links ] Received : April 2004. Accepted : October 2004 Ademir F. Pazoto Lucicléia Coelho and Ruy Coimbra Charao |
