Asymptotic Behaviour of a Viscoelastic Transmission Problem with a Tip Load

Míriam S. Carneiro, Marco A. A. Fernandes, Jaime E. Muñoz Rivera

Resumo


We consider a transmission problem for a string composed by two components: one ofthem is a viscoelastic material (with viscoelasticity of memory type), and the other is an elasticmaterial (without dissipation effective over this component). Additionally, we consider that in oneend is attached a tip load. The main result is that the model is exponentially stable if and only ifthe memory effect is effective over the string. When there is no memory effect, then there is a lackof exponential stability, but the tip load produces a polynomial rate of decay. That is, the tip loadis not strong enough to stabilize exponentially the system, but produces a polynomial rate of decay. 

Palavras-chave


Transmission problems; memory effect; lack of exponential stability; tip load; hybridsystem

Texto completo:

PDF (English)

Referências


Andrews, K.T. and Shillor, M.Vibrations of a Beam With a Damping Tip Body. Mathematicaland Computer Modelling 35 (2002), 1033-1042.

Borichev, A. and Tomilov, Y.Optimal Polynomial Decay of F

unctions and Operator Semi-groups. Mathematische Annalen. Vol. 347 (2), (2009), 455-478.

Dafermos, C. M.On Abstract Volterra Equation with Applications to Linear Viscoleasticity.Differential and Integral Equations. Vol. 7(1) (1970), 554-569.

Dafermos, C. M.Asymptotic Stability in Viscoleasticity. Arch. Rat. Mech. Anal.. Vol. 37(1)(1970), 297-308.

Engel, K. J. and Nagel, R.One-parameter Semigroups for Linear Evolution Equations.Grad-uate Texts in Mathematics (With contributions by S. Brendle, M. Campiti, T. Hahn, G.Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt).Springer-Verlag (2000).

Fabrizio, M. and Morro, A.Mathematical Problems in Linear Viscoleasticity. SIAM Studiesin Applied Mathematics. Vol. 12 (1992).[7] Feireisl, Eduard and O’Dowd, Geoffrey.Stabilisation d’un syst`eme hybride par un feedback nonlin ́eaire, non monotone. Comptes Rendus de l’Acad ́emie des Sciences, S ́erie I, Math ́ematique,Vol. 326(3), (1998), 323-327.

Kato, T.Perturbation Theory for Linear Operators. Springer-Verlag, New York, (1980).[9] Pr ̈uss, J.On the Spectrum ofC0-semigroups. Trans. AMS. 284 (1984), 847-857.




DOI: https://doi.org/10.5540/03.2021.008.01.0375

Apontamentos

  • Não há apontamentos.


SBMAC - Sociedade de Matemática Aplicada e Computacional
Edifício Medical Center - Rua Maestro João Seppe, nº. 900, 16º. andar - Sala 163 | São Carlos/SP - CEP: 13561-120
 


Normas para publicação | Contato