Energy decay rates for solutions of Maxwell's system with a memory boundary condition

• Autores: Serge Nicaise, Cristina Pignotti
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 58, Fasc. 3, 2007, págs. 327-342
• Idioma: español
• Enlaces
  • Texto completo
• Resumen

  • We consider the stabilization of Maxwell¿s equations with space variable coefficients in a bounded region with a smooth boundary, subject to dissipative boundary conditions of memory type on the boundary. Under suitable conditions on the domain and on the permeability and permittivity coefficients, we prove the exponential/polynomial decay of the energy. Our result is mainly based on the use of the multipliers method and the introduction of a suitable Lyapounov functional.

• Referencias bibliográficas
  • M. Aassila, M.M. Cavalcanti, and J.A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory...
  • D. Andrade and J.E. Muñoz Rivera, Exponential decay of non-linear wave equation with a viscoelastic boundary condition, Math. Methods Appl....
  • M.M. Cavalcanti, V.N. Domingos Cavalcanti, and M.L. Santos, Uniform decay rates of solutions to a nonlinear wave equation with boundary condition...
  • M.M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Differential...
  • M.M. Cavalcanti and H.P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42 (2003),...
  • G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl. (9)...
  • M. Eller, J.E. Lagnese, and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comput. Appl. Math....
  • M. Fabrizio and A. Morro, A boundary condition with memory in electromagnetism, Arch. Rational Mech. Anal. 136 (1996), 359–381.
  • A. Guesmia, Stabilisation de l’équation des ondes avec conditions aux limites de type mémoire, Afrika Math. 10 (1999), 14–25.
  • A. Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep. 3 (1987), i–xxiv and 1–281.
  • A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal. 100 (1988), 191–206.
  • B.V. Kapitonov, Stabilization and exact boundary controllability for Maxwell’s equations, SIAM J. Control Optim. 32 (1994), 408–420.
  • B.V. Kapitonov and G.P. Menzala, Uniform stabilization for Maxwell’s equations with boundary conditions with memory, Asymptot. Anal. 26 (2001),...
  • M. Kirane and N.-E. Tatar, A memory type boundary stabilization of a mildly damped wave equation, E. J. Qual. Theory Differ. Equ., 6 (1999),...
  • M. Kirane and N.-E. Tatar, Non-existence results for a semilinear hyperbolic problem with boundary condition of memory type, Z. Anal. Anwendungen...
  • V. Komornik, Boundary stabilization, observation and control of Maxwell’s equations, Panamer. Math. J. 4 (1994), 47–61.
  • V. Komornik, Exact controllability and stabilization. The multiplier method, Masson, Paris; John Wiley and Sons, Ltd., Chichester, 1994.
  • V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. 69 (1990), 33–54.
  • F. Macià and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach, Asymptot. Anal. 32 (2002), 1–26.
  • R. Nibbi and S. Polidoro, Exponential decay for Maxwell’s equations with a boundary memory condition, J. Math. Anal. Appl. 302 (2005), 30–55.
  • S. Nicaise and C. Pignotti, Boundary stabilization of Maxwell’s equations with space-time variable coefficients, ESAIM Control Optim. Calc....
  • S. Nicaise and C. Pignotti, Internal and boundary observability estimates for heterogeneous Maxwell’s system, Appl. Math. Optim. 54 (2006),...
  • N.D. Phung, Contrôle et stabilisation d’ondes électromagnétiques, ESAIM Control Optim. Calc. Var. 5 (2000), 87–137 (electronic).
  • C. Pignotti, Observability and controllability of Maxwell’s equations, Rend. Mat. Appl. 19 (1999), 523–546.
  • T.H. Qin, Global solvability of nonlinear wave equation with a viscoelastic boundary condition, Chinese Ann. Math. Ser. B 14 (1993), 335–346.
  • M.L. Santos, Decay rates for solutions of a system of wave equations with memory, Electron. J. Differential Equations 73 (2001), 1–11.
  • M.L. Santos, Decay rates for solutions of a system of wave equations with memory, Electron. J. Differential Equations 38 (2002), 1–17.
  • P.F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim....
  • E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems, Asymptotic Anal. 1 (1988), 161–185.
  • E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim. 28 (1990), 466–477.