Viscoelastic versus frictional dissipation in a variable coefficients plate system with time-varying delay

  • Peipei Wang
  • Jianghao HaoEmail author


In this paper, we are concerned with variable coefficients plate system subjected to three partially distributed feedbacks: time-varying delay, frictional and viscoelastic dissipations. This work is devoted to, without any prior quantification of both decay rate of relaxation function and growth rate of frictional dissipation near the origin, establish a general decay result which corresponds to a certainly stable ODE. Our result extends the decay result obtained for some kind of problems with finite history to problem with infinite history. Moreover, this paper allows a wider class of kernels of infinite history, and the usual exponential and polynomial decay rates are only special cases. The proof is based on the multiplier method and some techniques about convex functionals.


Frictional damping Infinite history Partially distributed feedbacks Time-varying delay Variable coefficients 

Mathematics Subject Classification

35B37 35L55 74D05 93D15 



The authors cordially thank the anonymous referees for the valuable comments and suggestions which lead to the improvement of this paper. This work was partially supported by NNSF of China (11871315, 61374089), Natural Science Foundation of Shanxi Province of China (201801D121003) and Graduate student education innovation project of Shanxi Province of China (2018BY025).


  1. 1.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)CrossRefGoogle Scholar
  2. 2.
    Berrimi, S., Messaoudi, S.A.: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron. J. Differ. Equ. 88, 1–10 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cao, X.: Energy decay of solutions for a variable-coefficient viscoelastic wave equation with a weak nonlinear dissipation. J. Math. Phys. 57, 021509 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cavalcanti, M.M., Domingos Cavalcanti, V.N., Lasiecka, I., Falcão Nascimento, F.A.: Intrinstic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete Contin. Dyn. Syst. Ser. B 19(7), 1987–2012 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 2002(44), 1–14 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cavalcanti, M.M., Oquendo, H.P.: Frctional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42, 1310–1324 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chepyzhov, V.V., Pata, V.: Some remarks on stability of semigroups arising from linear viscoelasticity. Asymptot. Anal. 46, 251–273 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Datko, R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim. 26, 697–713 (1988)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Datko, R., Lagnese, J., Polis, M.P.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24, 152–156 (1986)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Feng, B.: Well-posedness and exponential stability for a plate equation with time-varying delay and past history. Z. Angew. Math. Phys. 68, 1–24 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Feng, S., Feng, D.: Nonlinear internal damping of wave equations with variable coefficients. Acta Math. Sin. 20, 1057–1072 (2004). English seriesMathSciNetCrossRefGoogle Scholar
  13. 13.
    Guesmia, A., Messaoudi, S.A.: A new approach to the stability of an abstract system in the presence of infinite history. J. Math. Anal. Appl. 416, 212–228 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Guzmán, R.B., Tucsnak, M.: Energy decay estimates for the damped plate equation with a local degenerated dissipation. Syst. Control Lett. 48, 191–197 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lagnese, J.E.: Asymptotic energy estimates for Kirchoff plates subject to weak viscoelastic damping. Intern. Ser. Numer. Math. 91, 211–236 (1989)Google Scholar
  16. 16.
    Li, J., Chai, S.: Existence and energy decay rates of solutions to the variable-coefficient Euler–Bernoulli plate with a delay in localized nonlinear internal feedback. J. Math. Anal. Appl. 443, 981–1006 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integral Equ. 6, 507–533 (1993)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Li, S., Yao, P.: Stabilization of the Euler–Bernoulli plate with variable coefficients by nonlinear internal feedback. Automatica 50, 2225–2233 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mustafa, M.I.: Viscoelastic plate equation with boundary feedback. Evol. Equ. Control Theory 6, 261–276 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mustafa, M.I.: Uniform decay rates for viscoelastic dissipative systems. J. Dyn. Control Syst. 22, 101–116 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Martinez, P.: A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complut. 12, 251–283 (1999)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Martinez, P.: A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim. Calc. Var. 4, 419–444 (1999)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Muñoz Rivera, J.E., Barbosa Sobrinho, J.: Existence and uniform rates of decay for contact problems in viscoelasticity. Appl. Anal. 67, 175–199 (1997)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mustafa, M.I., Kafini, M.: Energy decay for viscoelastic plates with distributed delay and source term. Z. Angew. Math. Phys. 67, 1–18 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Muñoz Rivera, J.E., Lapa, E.C., Barreto, R.: Decay rates for viscoelastic plates with memory. J. Elast. 44, 61–87 (1996)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mustafa, M.I., Messaoudi, S.A.: General stability result for viscoelastic wave equations. J. Math. Phys. 53, 053702 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Muñoz Rivera, J.E., Peres Salvatierra, A.: Asymptotic behaviour of the energy in partially viscoelastic materials. Quart. Appl. Math. LIX(3), 557–578 (2001)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Nakao, M.: Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305, 403–417 (1996)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Renardy, M., Hrusa, W.J., Nohel, J.A.: Mathematical Problems in Viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow (1987)Google Scholar
  30. 30.
    Wang, P., Hao, J.: Asymptotic stability of memory-type Euler–Bernoulli plate with variable coefficients and time delay. J. Syst. Sci. Complex. (2019)
  31. 31.
    Wang, P., Hao, J.: General stability of a viscoelastic variable coefficients plate equation with time-varying delay in localized nonlinear internal feedback. J. Math. Anal. Appl. 466, 1582–1608 (2018)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yao, P.: On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J. Control Optim. 37, 1568–1599 (1999)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Yang, Z.: Existence and energy decay of solutions for the Euler–Bernoulli viscoelastic equation with a delay. Z. Angew. Math. Phys. 66, 727–745 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zuazua, E.: Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim. 28, 466–477 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanxi UniversityTaiyuanChina

Personalised recommendations