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General and Optimal Decay Result for a Viscoelastic Problem with Nonlinear Boundary Feedback

  • Mohammad M. Al-Gharabli
  • Adel M. Al-Mahdi
  • Salim A. Messaoudi
Article
  • 42 Downloads

Abstract

In this paper, we consider a viscoleastic equation with a nonlinear feedback localized on a part of the boundary and a relaxation function satisfying g(t) ≤−ξ(t)G(g(t)). We establish an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. Our results are obtained without imposing any restrictive growth assumption on the damping term. This work generalizes and improves earlier results in the literature, in particular those of Messaoudi (Topological Methods in Nonlinear Analysis 51(2):413–427, 2018), Messaoudi and Mustafa (Nonlinear Analysis: Theory Methods & Applications 72(9–10):3602–3611, 2010), Mustafa (Mathematical Methods in the Applied Sciences 41(1): 192–204, 2018) and Wu (Zeitschrift für angewandte Mathematik und Physik 63(1):65–106, 2012).

Keywords

Viscoelasticity Optimal decay Relaxation functions Convexity 

Mathematics Subject Classification (2010)

35B35 35L55 75D05 74D10 93D20 

Notes

Acknowledgments

The authors thank KFUPM for its continuous support. This work was funded by KFUPM under Project #IN161006.

Funding Information

This work was funded by KFUPM under Project #IN161006.

References

  1. 1.
    Messaoudi SA. Al-khulaifi General and optimal decay for a viscoelastic equation with boundary feedback. Topological Methods in Nonlinear Analysis 2018;51(2):413–427.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Messaoudi SA, Mustafa MI. On convexity for energy decay rates of a viscoelastic equation with boundary feedback. Nonlinear Analysis: Theory Methods & Applications 2010;72(9-10):3602–3611.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Mustafa MI. Optimal decay rates for the viscoelastic wave equation. Mathematical Methods in the Applied Sciences 2018;41(1):192–204.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Wu S-T. General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions. Zeitschrift für angewandte Mathematik und Physik 2012;63(1):65–106.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dafermos CM. Asymptotic stability in viscoelasticity. Archive for rational mechanics and analysis 1970;37(4):297–308.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dafermos CM. An, abstract volterra equation with applications to linear viscoelasticity. Journal of Differential Equations 1970;7(3):554–569.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hrusa WJ. Global existence and asymptotic stability for a semilinear hyperbolic volterra equation with large initial data. SIAM journal on mathematical analysis 1985; 16(1):110–134.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dassios G, Zafiropoulos F. Equipartition of energy in linearized 3-d viscoelasticity. Q Appl Math 1990;48(4):715–730.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Fabrizio M, Morro A. 1992. Mathematical problems in linear viscoelasticity, vol. 12. Siam.Google Scholar
  10. 10.
    Messaoudi SA. General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Analysis: Theory Methods & Applications 2008;69(8): 2589–2598.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Messaoudi SA. General decay of solutions of a viscoelastic equation. J Math Anal Appl 2008;341(2):1457–1467.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Han X, Wang M. General decay of energy for a viscoelastic equation with nonlinear damping. Mathematical Methods in the Applied Sciences 2009;32(3):346–358.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Liu W. 2009. General decay of solutions to a viscoelastic wave equation with nonlinear localized damping. In: Annales academiScientiarum fennicæ. Mathematica, vol 34, pp 291–302.Google Scholar
  14. 14.
    Liu W. General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. Journal of Mathematical Physics 2009;50 (11):113506.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Messaoudi SA, Mustafa MI. On the control of solutions of viscoelastic equations with boundary feedback. Nonlinear Analysis: Real World Applications 2009;10(5):3132–3140.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Mustafa MI. Uniform decay rates for viscoelastic dissipative systems. J Dyn Control Syst 2016;22(1):101–116.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Mustafa MI. Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations. Nonlinear Analysis: Real World Applications 2012;13(1): 452–463.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Park JY, Park SH. General decay for quasilinear viscoelastic equations with nonlinear weak damping. Journal of Mathematical Physics 2009;50(8):083505.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Wu S-T. General decay for a wave equation of kirchhoff type with a boundary control of memory type. Boundary Value Problems 2011;2011(1):55.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Lasiecka I, Tataru D, et al. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential Integral Equations 1993; 6(3):507–533.MathSciNetzbMATHGoogle Scholar
  21. 21.
    Alabau-Boussouira F, Cannarsa P. A general method for proving sharp energy decay rates for memory-dissipative evolution equations. Comptes Rendus Mathematiqué, 2009;347(15-16):867–872.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Cavalcanti MM, Domingos Cavalcanti VN, Lasiecka I, Falcao Nascimento FA. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete & Continuous Dynamical Systems-Series B 2014;19 (7):1987–2011.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Cavalcanti MM, Cavalcanti AD, Lasiecka I, Wang X. Existence and sharp decay rate estimates for a von karman system with long memory. Nonlinear Analysis: Real World Applications 2015;22:289–306.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Guesmia A. Asymptotic stability of abstract dissipative systems with infinite memory. J Math Anal Appl 2011;382(2):748–760.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Lasiecka I, Messaoudi SA, Mustafa MI. Note on intrinsic decay rates for abstract wave equations with memory. Journal of Mathematical Physics 2013;54(3):031504.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lasiecka I, Wang X. 2014. Intrinsic decay rate estimates for semilinear abstract second order equations with memory. In: New prospects in direct, inverse and control problems for evolution equations, pp 271–303. Springer.Google Scholar
  27. 27.
    Mustafa MI. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems-A 2015;35(3):1179–1192.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Xiao T-J, Liang J. Coupled second order semilinear evolution equations indirectly damped via memory effects. Journal of Differential Equations 2013;254(5):2128–2157.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Mustafa MI, Messaoudi SA. General stability result for viscoelastic wave equations. Journal of Mathematical Physics 2012;53(5):053702.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Cavalcanti MM, Cavalcanti VND, Lasiecka I, Webler CM. Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density. Advances in Nonlinear Analysis 2017;6(2):121–145.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Messaoudi SA, Al-Khulaifi W. General and optimal decay for a quasilinear viscoelastic equation. Appl Math Lett 2017;66:16–22.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Cavalcanti M, Cavalcanti VD, Prates Filho J, Soriano J, et al. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differential and integral equations 2001;14(1):85–116.MathSciNetzbMATHGoogle Scholar
  33. 33.
    Cavalcanti M, Cavalcanti VD, Martinez P. General decay rate estimates for viscoelastic dissipative systems. Nonlinear Analysis: Theory Methods & Applications 2008; 68(1):177–193.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Alabau-Boussouira F. Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl Math Optim 2005;51(1):61–105.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Cavalcanti MM, Cavalcanti VND, Lasiecka I. Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction. Journal of Differential Equations 2007;236(2):407–459.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Guesmia A. Well-posedness and optimal decay rates for the viscoelastic kirchhoff equation. Boletim da Sociedade Paranaense de Matematicá, 2017;35(3):203–224.MathSciNetCrossRefGoogle Scholar
  37. 37.
    Guesmia A. A new approach of stabilization of nondissipative distributed systems. SIAM journal on control and optimization 2003;42(1):24–52.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Cavalcanti M, Guesmia A, et al. General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type. Differential and Integral equations 2005;18(5):583–600.MathSciNetzbMATHGoogle Scholar
  39. 39.
    Berrimi S, Messaoudi SA. 2004. Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping., Electronic Journal of Differential Equations (EJDE)[electronic only], vol. 2004, pp Paper–No.Google Scholar
  40. 40.
    Komornik V. 1994. Exact controllability and stabilization: the multiplier method, vol. 36. Masson.Google Scholar
  41. 41.
    Arnol’d VI. 2013. Mathematical methods of classical mechanics, vol. 60. Springer Science & Business Media.Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.The Preparatory Year ProgramKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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