Journal of Integral Equations and Applications

Energy decay rates for solutions of the Kirchhoff type wave equation with boundary damping and source terms

Tae Gab Ha

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this work, we are concerned with uniform stabilization for an initial-boundary value problem associated with the Kirchhoff type wave equation with feedback terms and memory condition at the boundary. We prove that the energy decays exponentially when the boundary damping term has a linear growth near zero and polynomially when the boundary damping term has a polynomial growth near zero. Furthermore, we study the decay rate of the energy without imposing any restrictive growth assumption on the damping term near zero.

Article information

Source
J. Integral Equations Applications, Volume 30, Number 3 (2018), 377-415.

Dates
First available in Project Euclid: 8 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1541668119

Digital Object Identifier
doi:10.1216/JIE-2018-30-3-377

Mathematical Reviews number (MathSciNet)
MR3874007

Zentralblatt MATH identifier
06979946

Subjects
Primary: 35B35: Stability 35B40: Asymptotic behavior of solutions 35L05: Wave equation 35L20: Initial-boundary value problems for second-order hyperbolic equations

Keywords
Kirchhoff type wave equation energy decay rates

Citation

Ha, Tae Gab. Energy decay rates for solutions of the Kirchhoff type wave equation with boundary damping and source terms. J. Integral Equations Applications 30 (2018), no. 3, 377--415. doi:10.1216/JIE-2018-30-3-377. https://projecteuclid.org/euclid.jiea/1541668119


Export citation

References

  • 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. Contr. Optim. 38 (2000), 1581–1602.
  • F. Alabau-Boussouira, J. Prüss and R. Zacher, Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels, C.R. Acad. Sci. Paris 347 (2009), 277–282.
  • G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlin. Anal. 73 (2010), 1952–1965.
  • ––––, Kirchhoff systems with nonlinear source and boundary damping terms, Comm. Pure Appl. Anal. 9 (2010), 1161–1188.
  • J.J. Bae, Global existence and decay for Kirchhoff type wave equation with boundary and localized dissipations in exterior domains, Funk. Ekvac. 47 (2004), 453–477.
  • J.J. Bae and M. Nakao, Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains, Discr. Cont. Dynam. Syst. 11 (2004), 731–743.
  • M.M. Cavalcanti, V.N. Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Diff. Eqs. 203 (2004), 119–158.
  • E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerch. Mat. 8 (1959), 24–51.
  • G.C. Gorain, Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in $\mathbb{R}^n$, J. Math. Anal. Appl. 319 (2006), 635–650.
  • ––––, Exponential energy decay estimates for the solutions of $n$-dimensional Kirchhoff type wave equation, Appl. Math. Comp. 177 (2006), 235–242.
  • T.G. Ha, Asymptotic stability of the semilinear wave equation with boundary damping and source term, C.R. Math. Acad. Sci. Paris 352 (2014), 213–218.
  • ––––, General decay estimates for the wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary, Appl. Math. Lett. 60 (2016), 43–49.
  • ––––, General decay rate estimates for viscoelastic wave equation with Balakrishnan-Taylor damping, Z. Angew. Math. Phys. 67 (2016), Art. 32.
  • ––––, Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions, Discr. Cont. Dynam. Syst. 36 (2016), 6899–6919.
  • ––––, Global existence and uniform decay of coupled wave equation of Kirchhoff type in a noncylindrical domain, J. Korean Math. Soc. 54 (2017), 1081–1097.
  • ––––, On viscoelastic wave equation with nonlinear boundary damping and source term, Comm. Pur. Appl. Anal. 9 (2010), 1543–1576.
  • ––––, Stabilization for the wave equation with variable coefficients and Balakrishnan-Taylor damping, Taiwanese J. Math. 21 (2017), 807–817.
  • T.G. Ha, D. Kim and I.H. Jung, Global existence and uniform decay rates for the semi-linear wave equation with damping and source terms, Comp. Math. Appl. 67 (2014), 692–707.
  • T.G. Ha and J.Y. Park, Global existence and uniform decay of a damped Klein-Gordon equation in a noncylindrical domain, Nonlin. Anal. 74 (2011), 577–584.
  • T.G. Ha and J.Y. Park, Existence of solutions for the Kirchhoff type wave equation with memory term and acoustic boundary conditions, Numer. Funct. Anal. Optim. 31 (2010), 921–935.
  • ––––, Stabilization of the viscoelastic Euler-Bernoulli type equation with a local nonlinear dissipation, Dynam. PDE 6 (2009), 335–366.
  • H. Harrison, Plane and circular motion of a string, J. Acoust. Soc. Amer. 20 (1948), 874–875.
  • V. Komornik, Exact Controllability And Stabilization. The multiplier method, John Wiley, Paris, 1994.
  • ––––, On the nonlinear boundary stabilization of the wave equation, Chinese Ann. Math. 14 (1993), 153–164.
  • V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pure Appl. 69 (1990), 33–54.
  • J. Lagnese, Decay of solutions of the wave equation in a bounded region with boundary dissipation, J. Diff. Eqs. 50 (1983), 163–182.
  • I. Lasiecka and J. Ong, Global solvability and uniform decays of solution to quasilinear equation with nonlinear boundary dissipation, Comm. PDE 24 (1999), 2069–2107.
  • I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Diff. Int. Eqs. 6 (1993), 507–533.
  • P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM: Contr. Optim. Calc. Var. 4 (1999), 419–444.
  • A.H. Nayfeh and D.T. Mook, Nonlinear oscillations, Wiley, New York, 1979.
  • S. Nicaise and C. Pignotti, Stabilization of the wave equation with variable coefficients and boundary condition of memory type, Asympt. Anal. 50 (2006), 31–67.
  • L. Nirenberg, On elliptic partial differential equations, Ann. Scuol. Norm. Pisa 13 (1959), 115–162.
  • J.Y. Park, J.J. Bae and I.H. Jung, Uniform decay of solution for wave equation of Kirchhoff type with nonlinear boundary damping and memory term, Nonlin. Anal. 50 (2002), 871–884.
  • J.Y. Park and T.G. Ha, Existence and asymptotic stability for the semilinear wave equation with boundary damping and source term, J. Math. Phys. 49 (2008), 053511.
  • ––––, Energy decay for nondissipative distributed systems with boundary damping and source term, Nonlin. Anal. 70 (2009), 2416–2434.
  • ––––, Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions, J. Math. Phys. 50 (2009), 013506.
  • J.Y. Park, T.G. Ha and Y.H. Kang, Energy decay rates for solutions of the wave equation with boundary damping and source term, Z. Angew. Math. Phys. 61 (2010), 235–265.
  • J. Prüss, Evolutionary integral equations and applications, Birkhäuser-Verlag, Basel, 1993.
  • A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl. 71 (1992), 455–467.
  • M.L. Santos, J. Ferreira, D.C. Pereira and C.A. Raposo, Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary, Nonlinear Anal. 54 (2003), 959–976.
  • R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach, J. Math. Anal. Appl. 137 (1989), 438–461.
  • E. Vitillaro, A potential well method for the wave equation with nonlinear source and boundary damping terms, Glasgow Math. J. 44 (2002), 375–395.
  • R. Zacher, Quasilinear parabolic integro-differential equations with nonlinear boundary conditions, Diff. Int. Eqs. 19 (2006), 1129–1156.
  • E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. PDE 15 (1990), 205–235.
  • ––––, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Contr. Optim. 28 (1990), 466–477.