Study of the regularity of the Kirchhoff plate with intermediate damping

• DOI: 10.22533/at.ed.2582315082

• Palavras-chave: -

• Keywords: Asymptotic behaviour, Stability, Regularity, Gevrey Sharp-class, Analyticity, Kirchhoff Plates.

• Abstract:

  In this work, we study the regularity of the Kirchhoff plate equation with
  intermediate damping. The intermediate damping is given by A θ u t , where u is
  the displacement of the plate, A θ = (−∆) θ is a strictly positive self-adjoint operator
  on a complex Hilbert space for any real value of the parameter θ, here we will
  consider θ ∈ [0, 2]. In 2020 Vila et al. [2], published the study of the
  polynomial decay of this model considering the parameter θ ∈ [0, 1). More
  recently, in 2021, Tebou [23] published a study of asymptotic behavior and
  regularity considering intermediary damping, but considering θ ∈ [0, 2] and

  1

  2θ− , it is also shown that when the parameter θ2

  also considering a new parameter that considers the plate equations in-
  termediate between the Euler-Bernoulli and Kirchoff plates. Our research, like
  the two previous ones, also uses the theory of semigroups for the exis- tence,
  asymptotic behavior, and regularity of the semigroup S(t) associated with the
  model, we use the good properties of the operator −∆ to perform a spectral
  analysis of the model and demonstrate our results in a direct and friendly way:
  We show that the semigroup S(t) associated with the model is exponentially
  stable for θ ∈ [1, 2], we address the study of the analytic of S(t) for θ ∈ [ , 2]
  and that S(t) is not analytic when θ ∈ [0, ). In the last part of
  our investigation we show that for 1 < θ < 3 , S(t) has Gevrey Sharp classes given
  by s > 1 2 ∈ [0, 1] the
  semigroup S(t) does not admit Gevrey classes. For the study of existence,
  stability and regularity, semigroup theory is used together with frequency do-
  main techniques, multipliers, and spectral analysis of the fractional operator A θ
  for θ ∈ [0, 2] and inequality interpolation.