Decay rate estimates for a new class of multidimensional nonlinear Bresse systems with time-dependent dissipations

Abstract

This paper studies the existence of weak solutions and their asymptotic behaviour to the initial boundary value problem for a multidimensional nonlinear Bresse system. The existence of the solutions of the problem is obtained by appliying Galerkin method. Then, we obtain an explicit decay rate estimation dependent on both the strain-caused stress and the damping terms by using the multiplier method with integral.

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References

  1. 1.

    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14(4), 349–381 (1973)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Andrews, G., Ball, J.M.: Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity. J. Differ. Equ. 44(2), 306–341 (1982)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Boussouira, F.A., Rivera, J.E.M., da S Almeida Júnior, D.: Stability to weak dissipative Bresse system. J. Math. Anal. Appl. 374(2), 481–498 (2011)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Charles, W., Soriano, J.A., Nascimento, F.A.F., Rodrigues, J.H.: Decay rates for Bresse system with arbitrary nonlinear localized damping. J. Differ. Equ. 255(8), 2267–2290 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Dell’Oro, F.: Asymptotic stability of thermoelastic systems of Bresse type. J. Differ. Equ. 258(11), 3902–3927 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Dridi, H., Djebabla, A.: On the stabilization of linear porous elastic materials by microtemperature effect and porous damping. Ann. Univ. Ferrara 66, 13–25 (2020)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dridi, H., Djebabla, A.: Timoshenko system with fractional operator in the memory and spatial fractional thermal effect. Rend. Circ. Mat. Palermo II. Ser (2020). https://doi.org/10.1007/s12215-020-00513-6

    Article  MATH  Google Scholar 

  8. 8.

    Fatori, L.H., Monteiro, R.N.: The optimal decay rate for a weak dissipative Bresse system. Appl. Math. Lett. 25(3), 600–604 (2012)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Fatori, L.H., Rivera, J.E.M.: Rates of decay to weak thermoelastic Bresse system. IMA J. Appl. Math. 75(6), 881–904 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Green, A.E., Naghdi, P.M.: Errata: a re-examination of the basic postulates of thermomechanics. RSPSA 438(1904), 605 (1992)

    Google Scholar 

  11. 11.

    Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31(3), 189–208 (1993)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Guesmia, A.: Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements. Mediterr. J. Math. 14(2), 49 (2017)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Guesmia, A., Kafini, M.: Bresse system with infinite memories. Math. Methods Appl. Sci. 38(11), 2389–2402 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Guesmia, A., Kirane, M.: Uniform and weak stability of Bresse system with two infinite memories. Z. Angew. Math. Phys. 67(5), 124 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Keddi, A.A., Apalara, T.A., Messaoudi, S.A.: Exponential and polynomial decay in a thermoelastic-Bresse system with second sound. Appl. Math. Optim. 77(2), 315–341 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Khemmoudj, A.: General decay of solutions of a thermoelastic Bresse system with viscoelastic boundary conditions. Bol. Soc. Parana. Mat. 39(6), 157–182 (2020)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Lagnese, J.E., Leugering, G., Schmidt, E.J.P.G.: Modelling of dynamic networks of thin thermoelastic beams. Math. Methods Appl. Sci 16, 327 (1993)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Lagnese, J.E., Leugering, G., Schmidt, E.J.P.G.: Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures. Springer, Berlin (2012)

    Google Scholar 

  19. 19.

    Martinez, P.: A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim. Calc. Var. 4, 419–444 (1999)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Rifo, S., Villagran, O.V., Rivera, J.E.M.: The lack of exponential stability of the hybrid Bresse system. J. Math. Anal. Appl. 436(1), 1–15 (2016)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Santos, M.L., Soufyane, A., Junior, D.S.A.: Asymptotic behavior to Bresse system with past history. Quart. Appl. Math 73(1), 23–54 (2015)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Soriano, J.A., Charles, W., Schulz, R.: Asymptotic stability for Bresse systems. J. Math. Anal. Appl. 412(1), 369–380 (2014)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Soriano, J.A., Rivera, J.E.M., Fatori, L.H.: Bresse system with indefinite damping. J. Math. Anal. Appl. 387(1), 284–290 (2012)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Timoshenko, S.P.: LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 41(245), 744–746 (1921)

    Article  Google Scholar 

  25. 25.

    Wirth, J.: Solution representations for a wave equation with weak dissipation. Math. Methods Appl. Sci. 27(1), 101–124 (2004)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Zhijian, Y.: Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms. Math. Methods Appl. Sci. 25(10), 795–814 (2002)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

This work was supported by the Directorate-General for Scientific Research and Technological Development (DGRSDT) (Algeria). The author would like to thank the editor and the referees for their very helpful suggestions.

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Correspondence to Hanni Dridi.

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Dridi, H. Decay rate estimates for a new class of multidimensional nonlinear Bresse systems with time-dependent dissipations. Ricerche mat (2021). https://doi.org/10.1007/s11587-020-00554-0

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Keywords

  • Multidimensional Bresse system
  • Nonlinear systems
  • Stability
  • Arch problems, damping terms
  • Galerkin method
  • Multiplier technique

Mathematics Subject Classification

  • 35B40
  • 45K05
  • 74D05
  • 74D99