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Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping

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Abstract

This work is concerned with a semilinear non-homogeneous Timoshenko system under the effect of two nonlinear localized frictional damping mechanisms. The main goal is to prove its uniform stability by imposing minimal amount of support for the damping and, as expected, without assuming any relation on the non-constant coefficients. This fact generalizes substantially the previous papers by Cavalcanti et al. (Z Angew Math Phys 65(6):1189–1206, 2014) and Santos et al. (Differ Integral Equ 27(1–2):1–26, 2014) at the levels of problem and method. It is worth mentioning that the methodologies of these latter cannot be applied to the semilinear case herein, namely when one considers the problem with nonlinear source terms. Thus, differently of Cavalcanti et al. (Z Angew Math Phys 65(6):1189–1206, 2014), Santos et al. (Differ Integral Equ 27(1–2):1–26, 2014), the proof of our main stability result relies on refined arguments of microlocal analysis due to Burq and Gérard (Contrôle Optimal des équations aux dérivées partielles, http://www.math.u-psud.fr/~burq/articles/coursX.pdf, 2001). As far as we know, it seems to be the first time that such a methodology has been employed to 1-D systems of Timoshenko type with nonlinear foundations.

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Notes

  1. One can use, for instance, the easy tool Wolfram. See on the website: https://www.wolframalpha.com/input/?i=3*(x(s))%5E2*(x%27(s))%5E2-2*s*x(s)*x%27(s)%2B4*(x(s))%5E2-s%5E2%3D0

References

  1. Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30(5), 1024–1065 (1992)

    Article  MathSciNet  Google Scholar 

  2. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff International Publishing, România, Bucuresti (1976)

    Book  Google Scholar 

  3. Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics, Springer, New York (2010)

    Book  Google Scholar 

  4. Brézis, H.: Operateurs Maximaux Monotones et Semigroups de Contractions dans les Spaces de Hilbert. North Holland Publishing Co., Amsterdam (1973)

    MATH  Google Scholar 

  5. Burq, N.: Mesures semi-classiques et mesures de défaut. (French) [Semiclassical measures and defect measures] Séminaire Bourbaki, Vol. 1996/97. Astérisque No. 245 (1997), Exp. No. 826, 4, 167-195

  6. Burq, N., Gérard, P.: Contrôle Optimal des équations aux dérivées partielles. 2001, http://www.math.u-psud.fr/~burq/articles/coursX.pdf

  7. Burq, N., Gérard, P.: Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. (French) [A necessary and sufficient condition for the exact controllability of the wave equation]. C. R. Acad. Sci. Paris Sér. I Math. 325(7), 749–752 (1997)

  8. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Falcão Nascimento, F.A., Lasiecka, I., Rodrigues, J.H.: Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping. Z. Angew. Math. Phys. 65(6), 1189–1206 (2014)

    Article  MathSciNet  Google Scholar 

  9. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Fukuoka, R., Toundykov, D.: Stabilization of the damped wave equation with Cauchy-Ventcel boundary conditions. J. Evol. Equ. 9(1), 143–169 (2009)

    Article  MathSciNet  Google Scholar 

  10. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Fukuoka, R., Soriano, J.A.: Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result. Trans. Am. Math. Soc. 361(9), 4561–4580 (2009)

    Article  MathSciNet  Google Scholar 

  11. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Fukuoka, R., Soriano, J.A.: Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result. Arch. Ration. Mech. Anal. 197(3), 925–964 (2010)

    Article  MathSciNet  Google Scholar 

  12. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Fukuoka, R., Pampu, A.B., Astudillo, M.: Uniform decay rate estimates for the semilinear wave equation in inhomogeneous medium with locally distributed nonlinear damping. Nonlinearity 31(9), 4031–4064 (2018)

    Article  MathSciNet  Google Scholar 

  13. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Lasiecka, I.: Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J. Differ. Equ. 236(2), 407–459 (2007)

    Article  MathSciNet  Google Scholar 

  14. Cavalcanti, M.M., Khemmoudj, A., Medjden, M.: Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions. J. Math. Anal. Appl. 328(2), 900–930 (2007)

    Article  MathSciNet  Google Scholar 

  15. Chueshov, I., Eller, M., Lasiecka, I.: On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Commun. Part. Differ. Equ. 27(9–10), 1901–1951 (2002)

    Article  MathSciNet  Google Scholar 

  16. Daloutli, M., Lasiecka, I., Toundykov, D.: Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discret. Contin. Dyn. Syst. 2, 67–94 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Dehman, B., Gérard, P., Lebeau, G.: Stabilization and control for the nonlinear Schrödinger equation on a compact surface. Math. Z. 254(4), 729–749 (2006)

    Article  MathSciNet  Google Scholar 

  18. Dehman, B., Lebeau, G., Zuazua, E.: Stabilization and control for the subcritical semilinear wave equation. Ann. Sci. École Norm. Sup. 36(4), 525–551 (2003)

    Article  MathSciNet  Google Scholar 

  19. Fatori, L.H., Jorge Silva, M.A., Narciso, V.: Quasi-stability property and attractors for a semilinear Timoshenko system. Discrete Contin. Dyn. Syst. 36(11), 6117–6132 (2016)

    Article  MathSciNet  Google Scholar 

  20. Gérard, P.: Microlocal defect measures. Commun. Partial Differ. Equ. 16, 1761–1794 (1991)

    Article  MathSciNet  Google Scholar 

  21. Gérard, P., Leichtnam, E.: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J71(2), 559–607 (1993)

    MathSciNet  MATH  Google Scholar 

  22. Koch, H., Tataru, D.: Dispersive estimates for principally normal pseudodifferential operators. Commun. Pure Appl. Math. 58(2), 217–284 (2005)

    Article  MathSciNet  Google Scholar 

  23. Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integral Equ. 6, 507–533 (1993)

    MathSciNet  MATH  Google Scholar 

  24. Lasiecka, I., Toundykov, D.: Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms. Nonlinear Anal. 64, 1757–1797 (2006)

    Article  MathSciNet  Google Scholar 

  25. Lasiecka, I., Triggiani, R.: \(L_2\) regularity of the boundary to boundary operators \( B^* L\) for hyperbolic and Petrovsky PDE’s. Abstract Appl. Anal. 19, 1061–1139 (2003)

  26. Léautaud, M., Lerner, N.: Energy decay for a locally undamped wave equation. Ann. Fac. Sci. Toulouse Math. 26(1), 157–205 (2017)

    Article  MathSciNet  Google Scholar 

  27. Ma, T.F., Monteiro, R.N., Pereira, A.C.: Pullback dynamics of non-autonomous Timoshenko systems. Appl. Math. Optim. 80, 391–413 (2019)

    Article  MathSciNet  Google Scholar 

  28. Ruiz, A.: Unique continuation for weak solutions of the wave equation plus a Potential. J. Math. Pures Appl. 71,(1992)

  29. Raposo, C.A., Ferreira, J., Santos, M.L., Castro, N.N.O.: Exponential stability for the Timoshenko system with two weak dampings. Appl. Math. Lett. 18(5), 535–541 (2005)

    Article  MathSciNet  Google Scholar 

  30. Santos, M.L., Almeida Júnior, D.S., Rodrigues, J.H., Falcão Nascimento, F.A.: Decay rates for Timoshenko system with nonlinear arbitrary localized damping. Differ. Integral Equ. 27(1–2), 1–26 (2014)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Professor Nicolas Burq for fruitful discussions regarding the bicharacteristic flow in 1-D. Moreover, the authors would like to express their gratitude to the anonymous referee for giving constructive and fruitful suggestions, which have allowed us to improve on the presentation of our work.

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Authors and Affiliations

  1. Department of Mathematics, State University of Maringá, Maringá, PR, 87020-900, Brazil

    M. M. Cavalcanti

  2. Federal Technological University of Paraná, Campus Campo Mourão, Campo Mourão, PR, 87301-899, Brazil

    W. J. Corrêa

  3. Departamento de Mathematics, State University of Maringá, Maringá, PR, 87020-900, Brazil

    V. N. Domingos Cavalcanti

  4. Department of Mathematics, State University of Londrina, Londrina, PR, 86057-970, Brazil

    M. A. Jorge Silva

  5. Department of Mathematics, State University of Maringá, Maringá, PR, 87020-900, Brazil

    J. P. Zanchetta

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  1. M. M. Cavalcanti
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  2. W. J. Corrêa
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  3. V. N. Domingos Cavalcanti
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  4. M. A. Jorge Silva
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  5. J. P. Zanchetta
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Correspondence to W. J. Corrêa.

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Research of Cavalcanti partially supported by the CNPq, Grant 300631/2003-0. Research of Corrêa partially supported by the CNPq, Grant 305192/2019-1. Research of Domingos Cavalcanti partially supported by the CNPq, Grant 304895/2003-2. Research of Jorge Silva partially supported by the CNPq, Grant 301116/2019-9. Research of Zanchetta partially supported by the CAPES, Finance Code 001.

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Cavalcanti, M.M., Corrêa, W.J., Cavalcanti, V.N.D. et al. Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping. Z. Angew. Math. Phys. 72, 191 (2021). https://doi.org/10.1007/s00033-021-01622-7

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