  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 the fractional coupling is given by: $\eta A^\phi\theta,
\alpha\eta A^\phi u_{tt}$ and $\eta A^\phi u_t$, where the operator $A^\phi$ is
self-adjoint and strictly positive in a complex Hilbert space $H$ and the
parameter $\phi$ can vary between $0$ and $1$. When $\phi=1$ we have the
MGT-Fourier physical model, previously investigated, see;
2013\cite{ABMvFJRSV2013} and 2022\cite{DellOroPata2022}, in these works the
authors respectively showed that the semigroup $S(t) = e^{t\mathbb{B}}$
associated with the MGT-Fourier model are exponentially stable and analytical.
The model abstract of this research is given by: \eqref{Eq1.1}--\eqref{Eq1.3},
we show directly that the semigroup $S(t)$ is exponentially stable for $\phi
\in [0,1]$, we also show that for $\phi=1$, $S(t)$ is analytic and study of the
Gevrey classes of $S(t)$ and we show that for $\phi\in (\frac{1}{2}, 1)$ there
are two families of Gevrey classes: $s_1>2$ when $\phi\in(1/2,2/3]$ and
$s_2>\frac{\phi}{2\phi-1}$ when $\phi\in[2/3,1)$, 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 $\phi \in[0,1/2]$.
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 system, using proprety of the fractional operator $A^\phi$
for $\phi\in[0,1]$.
