Advertisement

Optimal decay rates for the abstract viscoelastic equation

  • Muhammad I. MustafaEmail author
Article
  • 3 Downloads

Abstract

In this paper, we consider an abstract viscoelastic equation with minimal conditions on the \(L^{1}(0,\infty )\) relaxation function g namely \( g^{\prime }(t)\le -\xi (t)H(g(t))\), where H is an increasing and convex function near the origin and \(\xi \) is a nonincreasing function. With only these very general assumptions on the behavior of g at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when \(H(s)=s^{p}\) and p covers the full admissible range [1, 2). We get the best decay rates expected under this level of generality and our new results substantially improve several earlier related results in the literature.

Keywords

General decay Viscoelastic damping Relaxation function Convexity 

Mathematics Subject Classification

35B40 74D99 93D15 93D20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by MASEP Research Group in the Research Institute of Sciences and Engineering at University of Sharjah.

References

  1. 1.
    Alabau-Boussouira F. and Cannarsa P., A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris, Ser. I 347 (2009), 867–872.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alabau-Boussouira F., Cannarsa P. and Sforza D., Decay estimates for the second order evalution equation with memory, J. Funct. Anal. 245 (2008), 1342–1372.CrossRefzbMATHGoogle Scholar
  3. 3.
    Arnold V. I., Mathematical methods of classical mechanics, Springer-Verlag, New York, 1989.CrossRefGoogle Scholar
  4. 4.
    Cabanillas E.L. and Munoz Rivera J.E., Decay rates of solutions of an anisotropic inhomogeneous \(n\)-dimensional viscoelastic equation with polynomial decaying kernels, Comm. Math. Phys. 177 (1996), 583–602.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cannarsa P. and Sforza D., Integro-differential equations of hyperbolic type with positive definite kernels, J. Differential Equations 250 (2011), 4289–4335.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cavalcanti M.M., Cavalcanti V.N.D., Lasiecka I. and Nascimento F.A., Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Contin. Dyn. Syst. Ser. B, 19(7) (2014), 1987–2012.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cavalcanti M.M., Cavalcanti A.D.D., Lasiecka I. and Wang X., Existence and sharp decay rate estimates for a von Karman system with long memory, Nonlinear Anal. RWA 22 (2015), 289–306.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cavalcanti M.M. and Oquendo H.P., Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42(4) (2003), 1310–1324.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dafermos C.M., On abstract Volterra equations with applications to linear viscoelasticity, J. Differential Equations 7 (1970), 554–569.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dafermos C.M., Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297–308.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Guesmia A., Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl. 382 (2011), 748–760.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Han Xi. and Wang M., General decay of energy for a viscoelastic equation with nonlinear damping, Math. Meth. Appl. Sci. 32(3) (2009), 346–358.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hrusa W.J., Global existence and asymptotic stability for a semilinear Volterra equation with large initial data, SIAM Journal of Math. Anal. 16(1) (1985), 110–134.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jin K.P., Liang J. and Xiao T.J., Coupled second order evolution equations with fading memory: Optimal energy decay rate, J. Differential Equations 257 (2014), 1501–1528.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lasiecka I., Messaoudi S.A. and Mustafa M I., Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys. 54, 031504 (2013).  https://doi.org/10.1063/1.4793988.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lasiecka I. and Tataru D., Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations 8 (1993), 507–533.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Lasiecka I. and Wang X., Intrinsic decay rate estimates for semilinear abstract second order equations with memory, New prospects in direct, inverse and control problems for evolution equations, 271303, Springer INdAM Ser., 10, Springer, Cham, 2014.Google Scholar
  18. 18.
    Liu W.J., General decay of solutions to a viscoelastic wave equation with nonlinear localized damping. Ann. Acad. Sci. Fenn. Math. 34 no. 1 (2009), 291–302.MathSciNetzbMATHGoogle Scholar
  19. 19.
    Liu W.J., General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms, J. Math. Physics 50 (11), art. No 113506 (2009).Google Scholar
  20. 20.
    Messaoudi S.A. General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl. 341 (2008), 1457–1467.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Messaoudi S.A., General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal. 69 (2008), 2589–2598.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Messaoudi S.A. and Mustafa M.I., On the control of solutions of viscoelastic equations with boundary feedback, Nonlinear Anal.: RWA 10 (2009), 3132–3140.Google Scholar
  23. 23.
    Munoz Rivera J.E., Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math. 52(4) (1994), 628–648.CrossRefGoogle Scholar
  24. 24.
    Munoz Rivera J.E. and Peres Salvatierra A., Asymptotic behavior of the energy in partially viscoelastic materials, Quart. Appl. Math. 59(3) (2001), 557–578.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Munoz Rivera J.E., Naso M.G. and Vegni F.M., Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl. 286(2) (2003), 692–704.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Munoz Rivera J.E. and Naso M.G. , On the decay of the energy for systems with memory and indefinite dissipation, Asympt. Anal. 49(3-4) (2006), 189–204.MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mustafa M.I., On the control of the wave equation by memory-type boundary condition, Discrete Contin. Dyn. Syst. Ser. A 35(3) (2015), 1179–1192.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mustafa M.I., Uniform decay rates for viscoelastic dissipative systems, J. Dyn. Control Syst. 22 (1) (2016), 101–116.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mustafa M.I., Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations, Nonlinear Anal. RWA 13 (2012), 452–463.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mustafa M.I. and Messaoudi S.A., General stability result for viscoelastic wave equations, J. Math. Phys. 53 (2012), 053702.  https://doi.org/10.1063/1.4711830.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Park J. and Park S., General decay for quasilinear viscoelastic equations with nonlinear weak damping, J. Math. Physics 50 (8), art. No 083505 (2009).Google Scholar
  32. 32.
    Santos M.L., Asymptotic behavior of solutions to wave equations with a memory conditions at the boundary, Electron. J. Differ. Equ. 73 (2001), 1–11.MathSciNetGoogle Scholar
  33. 33.
    Wu S.T., General decay for a wave equation of Kirchho type with a boundary control of memory type, Boundary Value Problems 2011 (2011), 15pp.Google Scholar
  34. 34.
    Xiao T.J. and Liang J., Coupled second order semilinear evolution equations indirectly damped via memory effects, J. Differential Equations 254(5) (2013), 2128–2157.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SharjahSharjahUnited Arab Emirates

Personalised recommendations