The regularity of the coupled system between an electrical network with fractional dissipation and a plate equation with fractional inertial rotational

Abstract

In this work we study a strongly coupled system between the equation of plates with fractional rotational inertial force $\kappa (-\Delta {)}^{\beta }{u}_{tt}$ where the parameter $0<\beta \le 1$ and the equation of an electrical network containing a fractional dissipation term $\delta (-\Delta {)}^{\theta }{v}_t$ where the parameter $0\le \theta \le 1$, the strong coupling terms are given by the Laplacian of the displacement speed $\gamma \Delta u_t$ and the Laplacian electric potential field $\gamma \Delta v_t$. When $\beta =1$, we have the Kirchoff-Love plate and when $\beta =0$, we have the Euler-Bernoulli plate recently studied in Suárez-Mendes (2022-Preprinter)\cite{Suarez}. The contributions of this research are: We prove the semigroup $S(t)$ associated with the system is not analytic in $(\theta ,\beta )\in [0,1]\times (0,1]-\{(1,1/2)\}$. We also determine two Gevrey classes: $s_1>\frac{1}{2\max \{\frac{1-\beta }{3-\beta },\frac{\theta }{2+\theta -\beta }\}}$ for $2\le \theta +2\beta$ and $s_2>\frac{2(2+\theta -\beta )}{\theta }$ when the parameters $\theta$ and $\beta$ lies in the interval $(0,1)$ and we finish by proving that at the point $(\theta ,\beta )=(1,1/2)$ the semigroup $S(t)$ is analytic and with a note about the asymptotic behavior of $S(t)$. We apply semigroup theory, the frequency domain method together with multipliers and the proper decomposition of the system components and Lions' interpolation inequality.

40 pages