Taiwanese Journal of Mathematics

General Decay for a Viscoelastic Wave Equation with Density and Time Delay Term in $\mathbb{R}^n$

Baowei Feng

Full-text: Open access

Abstract

A linear viscoelastic wave equation with density and time delay in the whole space $\mathbb{R}^n$ ($n \geq 3$) is considered. In order to overcome the difficulties in the non-compactness of some operators, we introduce some weighted spaces. Under suitable assumptions on the relaxation function, we establish a general decay result of solution for the initial value problem by using energy perturbation method. Our result extends earlier results.

Article information

Source
Taiwanese J. Math., Volume 22, Number 1 (2018), 205-223.

Dates
Received: 16 November 2016
Revised: 27 April 2017
Accepted: 30 April 2017
First available in Project Euclid: 17 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1502935237

Digital Object Identifier
doi:10.11650/tjm/8105

Mathematical Reviews number (MathSciNet)
MR3749361

Zentralblatt MATH identifier
06965366

Subjects
Primary: 93D15: Stabilization of systems by feedback 35B40: Asymptotic behavior of solutions 35L05: Wave equation

Keywords
energy decay wave equation weighted space density delay feedbacks

Citation

Feng, Baowei. General Decay for a Viscoelastic Wave Equation with Density and Time Delay Term in $\mathbb{R}^n$. Taiwanese J. Math. 22 (2018), no. 1, 205--223. doi:10.11650/tjm/8105. https://projecteuclid.org/euclid.twjm/1502935237


Export citation

References

  • A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A 116 (1990), no. 3-4, 221–243.
  • X. Cao and P. Yao, General decay rate estimates for viscoelastic wave equation with variable coefficients, J. Syst. Sci. Complex. 27 (2014), no. 5, 836–852.
  • M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions, J. Math. Anal. Appl. 281 (2003), no. 1, 108–124.
  • Q. Dai and Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew Math. Phys. 65 (2014), no. 5, 885–903.
  • R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim. 24 (1986), no. 1, 152–156.
  • M. Kafini, Uniform decay of solutions to Cauchy viscoelastic problems with density, Elecron. J. Differential Equations 2011 (2011), no. 93, 9 pp.
  • M. Kafini and S. A. Messaoudi, A blow-up result in a Cauchy viscoelastic problem, Appl. Math. Lett. 21 (2008), no. 6, 549–553.
  • ––––, On the uniform decay in viscoelastic problem in $\mathbb{R}^n$, Appl. Math. Comput. 215 (2009), no. 3, 1161–1169.
  • M. Kafini, S. A. Messaoudi and S. Nicaise, A blow-up result in a nonlinear abstract evolution system with delay, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 2, Art. 13, 14 pp.
  • N. I. Karachalios and N. M. Stavrakakis, Existence of a global attractor for semilinear dissipative wave equations on $\mathbb{R}^n$, J. Differerential Equations 157 (1999), no. 1, 183–205.
  • M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys. 62 (2011), no. 6, 1065–1082.
  • W. Liu, General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback, J. Math. Phys. 54 (2013), no. 4, 043504, 9 pp.
  • S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl. 341 (2008), no. 2, 1457–1467.
  • ––––, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal. 69 (2008), no. 8, 2589–2598.
  • ––––, General decay of solutions of a weak viscoelastic equation, Arab. J. Sci. Eng. 36 (2011), no. 8, 1569–1579.
  • S. A. Messaoudi and M. M. Al-Gharabli, A general decay result of a nonlinear system of wave equations with infinite memories, Appl. Math. Comput. 259 (2015), 540–551.
  • S. A. Messaoudi and A. Soufyane, General decay of solutions of a wave equation with a boundary control of memory type, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2896–2904.
  • M. I. Mustafa and S. A. Messaoudi, General stability result for viscoelastic wave equations, J. Math. Phys. 53 (2012), no. 5, 053702, 14 pp.
  • S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim. 45 (2006), no. 5, 1561–1585.
  • ––––, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations 21 (2008), no. 9-10, 935–958.
  • ––––, Exponential stability of abstract evolution equations with time delay, J. Evol. Equ. 15 (2015), no. 1, 107–129.
  • S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control. Optim. Calc. Var. 16 (2010), no. 2, 420–456.
  • S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), no. 3, 559–581.
  • P. G. Papadopoulos and N. M. Stavrakakis, Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb{R}^n$, Topol. Methods Nonlinear Anal. 17 (2001), no. 1, 91–109.
  • B. Said-Houari, S. A. Messaoudi and A. Guesmia, General decay of solutions of a nonlinear system of viscoelastic wave equations, NoDEA Nonlinear Differential Equations Appl. 18 (2011), no. 6, 659–684.
  • N.-e. Tatar, Arbitrary decays in linear viscoelasticity, J. Math. Phys. 52 (2011), no. 1, 013502, 12 pp.
  • S.-T. Wu, General decay of solutions for a nonlinear system of viscoelastic wave equations with degenerate damping and source terms, J. Math. Anal. Appl. 406 (2013), no. 1, 34–48.
  • ––––, Asymptotic behavior for a viscoelastic wave equation with a delay term, Taiwanese J. Math. 17 (2013), no. 3, 765–784.
  • G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var. 12 (2006), no. 4, 770–785.
  • K. Zennir, General decay of solutions for damped wave equation of Kirchhoff type with density in $\mathbb{R}^n$, Ann. Univ. Ferrara Sez. VII Sci. Mat. 61 (2015), no. 2, 381–394.
  • Y. Zhou, A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in $\mathbb{R}^{N}$, Appl. Math. Lett. 18 (2005), no. 3, 281–286.
  • ––––, Global existence and nonexistence for a nonlinear wave equation with damping and source terms, Math. Nachr. 278 (2005), no. 11, 1341–1358.