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Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping

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Abstract

We consider the nonlinear viscoelastic equation

$$u_{tt}-\Delta u+\int_{0}^{t}g(t-\tau)\Delta u(\tau)\,d\tau +a(x)|u_{t}|^{m}u_{t}+b|u|^{\gamma }u=0$$

in a bounded domain and establish exponential or polynomial decay result which depend on the rate of the decay of the relaxation function g. This result improves an earlier one given by Berrimi and Messaoudi (Electron. J. Differ. Equ. (88):1–10, 2004).

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References

  1. Aassila, M., Cavalcanti, M.M., Soriano, J.A.: Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain. SIAM J. Control Optim. 38(5), 1581–1602 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alabau-Boussouira, F., Cannarsa, P., Sforza, D.: Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254(5), 1342–1372 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berrimi, S., Messaoudi, S.A.: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron. J. Differ. Equ. 2004(88), 1–10 (2004)

    MathSciNet  Google Scholar 

  4. Berrimi, S., Messaoudi, S.A.: Existence and decay of solutions of a viscoelastic equation with a localized damping and a nonlinear source. Nonlinear Anal. 64, 2314–2331 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cacalcanti, M.M., Oquendo, H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)

    Article  MathSciNet  Google Scholar 

  6. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Ferreira, J.: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24, 1043–1053 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Prates Filho, J.S., Soriano, J.A.: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ. Integral Equ. 14(1), 85–116 (2001)

    MATH  MathSciNet  Google Scholar 

  8. 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)

    MathSciNet  Google Scholar 

  9. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Martinez, P.: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. 68(1), 177–193 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Georgiev, V., Todorova, G.: Existence of solutions of the wave equation with nonlinear damping and source terms. J. Differ. Equ. 109, 295–308 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Haraux, A., Zuazua, E.: Decay estimates for some semilinear damped hyperbolic problems. Arch. Ration. Mech. Anal. 150, 191–206 (1988)

    Article  MathSciNet  Google Scholar 

  12. Kawashima, S., Shibata, Y.: Global existence and exponential stability of small solutions to nonlinear viscoelasticity. Commun. Math. Phys. 148(1), 189–208 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kopackova, M.: Remarks on bounded solutions of a semilinear dissipative hyperbolic equation. Comment. Math. Univ. Carol. 30, 713–719 (1989)

    MATH  MathSciNet  Google Scholar 

  14. Levine, H.A., Ro Park, S.: Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation. J. Math. Anal. Appl. 228, 181–205 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, W.J.: The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity. J. Math. Pures Appl. 37, 355–386 (1998)

    Google Scholar 

  16. Liu, W.J.: General decay of solutions to a viscoelastic equation with nonlinear localized damping, submitted

  17. Messaoudi, S.A., Tatar, N.-E.: Global existence asymptotic behavior for a nonlinear viscoelastic problem. Math. Meth. Sci. Res. J. 7(4), 136–149 (2003)

    MATH  MathSciNet  Google Scholar 

  18. Messaoudi, S.A.: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. 260, 58–66 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Messaoudiand, S.A., Tatar, N.-E.: Exponential and polynomial decay for a quasilinear viscoelastic equation. Nonlinear Anal. 68, 785–793 (2008)

    Article  MathSciNet  Google Scholar 

  20. Vitillaro, E.: Global nonexistence theorems for a class of evolution equations with dissipation. Arch. Ration. Mech. Anal. 149, 155–182 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Zuazua, E.: Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Differ. Equ. 15, 205–235 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Wenjun Liu

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  1. Wenjun Liu
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Correspondence to Wenjun Liu.

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Liu, W. Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping. J. Appl. Math. Comput. 32, 59–68 (2010). https://doi.org/10.1007/s12190-009-0232-y

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  • DOI: https://doi.org/10.1007/s12190-009-0232-y

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