Taiwanese Journal of Mathematics

GENERAL DECAY OF SOLUTIONS FOR A VISCOELASTIC EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING

Shun-Tang Wu

Full-text: Open access

Abstract

A viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary/interior sources is considered in a bounded domain. Under appropriate assumptions on the relaxation function and with certain initial data and by adopting the perturbed energy method, we establish uniform decay rate of the solution energy in terms of the behavior of the relaxation function, which are not necessarily of exponential or polynomial decay.

Article information

Source
Taiwanese J. Math., Volume 19, Number 2 (2015), 553-566.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133646

Digital Object Identifier
doi:10.11650/tjm.19.2015.4631

Mathematical Reviews number (MathSciNet)
MR3332313

Zentralblatt MATH identifier
1357.35229

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations 35L20: Initial-boundary value problems for second-order hyperbolic equations 58G16

Keywords
Balakrishnan-Taylor damping global existence general decay relaxation function viscoelastic equation

Citation

Wu, Shun-Tang. GENERAL DECAY OF SOLUTIONS FOR A VISCOELASTIC EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING. Taiwanese J. Math. 19 (2015), no. 2, 553--566. doi:10.11650/tjm.19.2015.4631. https://projecteuclid.org/euclid.twjm/1499133646


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References

  • F. Alabau-Boussouira, On convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.
  • A. V. Balakrishnan and L. W. Taylor, Distributed parameter nonlinear damping models for flight structures. in: Proceedings "Daming 89" , Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.
  • R. W. Bass and D. Zes, Spillover nonlinearity, and flexible structures. in: The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, Taylor, L. W. (ed.), NASA Conference Publication 10065, 1991, pp. 1-14.
  • M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Diff. Eqns., 236 (2007), 407-459.
  • M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., Theory, Methods & Applications, 68 (2008), 177-193.
  • M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J. A. Soriano, Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential and Integral Equations, 14 (2001), 85-116.
  • M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global solvability and asymptotic stability for the wave equation with nonlinear feedback and source term on the boundary, Advances in Mathematical Sciences and Applications, 16 (2006), 661-696.
  • H. R. Clark, Elastic membrane equation in bounded and unbounded domains, Electron. J. Qual. Theory Differ. Equ., 11 (2002), 1-21.
  • I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential and Integral Equations, 6 (1993), 507-533.
  • S. A. Messaoudi, General decay of solutions of a viscoelastic equation, Journal of Mathematical Analysis and Applications, 341 (2008), 1457-1467.
  • S. A. Messaoudi and M. I. Mustafa, On convexity for energy decay rates of a viscoelastic equation with boundary feedback, Nonlinear Anal., Theory, Methods & Applications, 72 (2010), 3602-3611.
  • C. L. Mu and J. Ma, On a system of nonlinear wave equations with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 65 (2014), 91-113.
  • J. E. Munoz Rivera, Global solution on a quasilinear wave equation with memory, Bolletino U.M.I., 7(8B) (1994), 289-303.
  • N.-e., Tatar, and, A., Zarai, Exponential, stability, and, blow, up, for, a, problem, with, Balakrishnan-Taylor damping, Demonstr. Math. XLIV, 1 (2011), 67-90.
  • R. M. Torrejón and J. Young, On a quasilinear wave equation with memory, Nonlinear Anal., Theory, Methods & Applications, 16 (1991), 61-78.
  • S. T. Wu, General decay and blow-up of solutions for a viscoelastic equation with a nonlinear boundary damping-source interactions, Z. Angew. Math. Phys., 63 (2012), 65-106.
  • Y. You, Intertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping, Abstr. Appl. Anal., 1 (1996), 83-102.
  • A. Zarai and N.-e. Tatar, Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping, Arch. Math. $($BRNO$)$, 46 (2010), 157-176.
  • A. Zarai, N.-e. Tatar and S. Abdelmalek, Elastic membrance equation with memory term and nonlinear boundary damping: global existence, decay and blowup of the solution, Acta Math. Sci., 33B (2013), 84-106.