Stability and Regularity the MGT-Fourier Model with Fractional Coupling

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 $\phi\in[0,1]$.