Asymptotic behavior for Timoshenko systems with fractional damping

Article type: Research Article

Authors: Oquendo, Higidio Portillo^{a; *} | da Luz, Cleverson Roberto^b

Affiliations: [a] Department of Mathematics, Federal University of Paraná, Brazil | [b] Department of Mathematics, Federal University of Santa Catarina, Brazil. E-mail: cleverson.luz@ufsc.br

Correspondence: [*] Corresponding author. E-mail: higidio@ufpr.br.

Abstract: This article deals with the asymptotic behavior of the solutions of a Timoshenko beam with a fractional damping. The damping acts only in one of the equations and depends on a parameter θ∈[0,1]. Timoshenko systems with frictional or Kelvin–Voigt dampings are particular cases of this model. We prove that, for regular initial data, the semigroup of this system decays polynomially with rates that depend on θ and some relations between the structural parameters of the system. We also show that the decay rates obtained are optimal and the only possibility to obtain exponential decay is when θ=0 and the wave propagation speeds of the equations coincide.

Keywords: Timoshenko beam, frictional damping, Kelvin–Voigt damping, polynomial decay, exponential decay

DOI: 10.3233/ASY-191552

Journal: Asymptotic Analysis, vol. 118, no. 1-2, pp. 123-142, 2020