Stability and Regularity the MGT-Fourier Model with Fractional Coupling

• DOI: 10.22533/at.ed.3742302082

• Palavras-chave: -

• Keywords: -

• Abstract:

  In this work, we study the stability and regularity of the system formed by
  the third-order vibration equation in Moore-Gilson-Thompson time coupled
  with the classical heat equation with Fourier’s law. We consider fractional
  couplings. He then fractional coupling is given by: ηAφ

  θ, αηAφutt and ηAφut
  ,

  where the operator Aφ

  is self-adjoint and strictly positive in a complex Hilbert
  space H and the parameter φ can vary between 0 and 1. When φ = 1 we have
  the MGT-Fourier physical model, previously investigated, see; 2013[1] and
  2022[9], in these works, the authors respectively showed that the semigroup
  S(t) = e
  tB associated with the MGT-Fourier model are exponentially stable
  and analytical. The model abstract of this research is given by: (3)–(5), we
  show directly that the semigroup S(t) is exponentially stable for φ ∈ [0, 1],
  we also show that for φ = 1, S(t) is analytic and study of the Gevrey classes
  of S(t). We show that for φ ∈ (
  1
  2
  , 1) there are two families of Gevrey classes:

  s1 > 2 when φ ∈ (1/2, 2/3] and s2 >
  φ
  2φ−1 when φ ∈ [2/3, 1), in the last part

  of our investigation using spectral analysis we tackled the study of the non-
  analyticity and lack of Gevrey classes of S(t) when φ ∈ [0, 1/2]. For the study

  of the existence, stability, and regularity, semigroup theory is used together
  with the techniques of the frequency domain, multipliers, and spectral analysis
  of a system, using the property of the fractional operator Aφ

  for φ ∈ [0, 1].