Advertisement

Uniform Decay Rates of Solutions to a Nonlinear Wave Equation with Boundary Condition of Memory Type

  • Marcelo M. Cavalcanti
  • Valéria N. Domingos Cavalcanti
  • Mauro L. Santos
Conference paper
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 166)

Abstract

In this article we study the hyperbolic problem (1) where Ω is a bounded region in R n whose boundary is partitioned into disjoint sets Γ0, Γ1. We prove that the dissipation given by the memory term is strong enough to assure exponential (or polynomial) decay provided the relaxation function also decays exponentially (or polynomially). In both cases the solution decays with the same rate of the relaxation function.

Keywords

wave equation gradient nonlinearity boundary memory term 

References

  1. [1]
    M. Aassila, M. M. Cavalcanti, and J. A. Soriano. Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain. SIAM J. Control Optim., 38(5):1581–1602, 2000.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    D. Andrade and J. E. Muñoz Rivera. Exponential decay of non-linear wave equation with viscoelastic boundary condition. Math. Meth. Appl. Sci, 23:41–61, 2000.MATHCrossRefGoogle Scholar
  3. [3]
    M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano. Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients. Electron. J. Differential Equations, 1998(08):1–21, 1998.MATHMathSciNetGoogle Scholar
  4. [4]
    M. Ciarletta. A differential problem for the heat equation with a boundary condition with memory. Appl. Math. Lett., 10(1):95–101, 1997.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Fabrizio and A. Morro. A boundary condition with memory in electromagnetism. Arch. Rational Mech. Anal., 136:359–381, 1996.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    M. Kirane and N. Tartar. Non-existence results for a semilinear hyperbolic problem with boundary condition of memory type. Journal for Analysis and Its Applications, 19(2):453–468, 2000.MATHMathSciNetGoogle Scholar
  7. [7]
    J. L. Lions. Quelques Mèthodes de resolution de problèmes aux limites non lineaires. Dunod Gauthiers Villars, Paris, 1969.MATHGoogle Scholar
  8. [8]
    T. Qin. Global solvability of nonlinear wave equation with a viscoelastic boundary condition. Chin. Ann. Math., 14B(3):335–346, 1993.Google Scholar
  9. [9]
    T. Qin. Breakdown of solutions to nonlinear wave equations with a viscoelastic boundary condition. Arab. J. Sci. Engng., 19(2A):195–201, 1994.MATHGoogle Scholar
  10. [10]
    R. Racke. Lectures on nonlinear evolution equations. Initial value problems. Aspect of Mathematics E19. Priedr. Vieweg & Sohn, Braunschweig, Wiesbaden, 1992.Google Scholar
  11. [11]
    M. L. Santos. Decay rates for solutions of a system of wave equations with memory. E. J. Diff. Eqs., 2002 (38):1–17.Google Scholar
  12. [12]
    M. L. Santos. Asymptotic behavior of solutions to wave equations with a memory condition at the boundary. E. J. Diff. Eqs., 2001 (73):1–11.MATHGoogle Scholar

Copyright information

© International Federation for Information Processing 2005

Authors and Affiliations

  • Marcelo M. Cavalcanti
    • 1
  • Valéria N. Domingos Cavalcanti
    • 1
  • Mauro L. Santos
    • 2
  1. 1.Department of MathematicsState University of MaringáMaringá - PRBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal do ParáParáBrazil

Personalised recommendations