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 $0 <\beta\leq 1$ and the equation of an electrical network containing a fractional dissipation term $\delta(-\Delta)^\theta v_t$ where the parameter $0\leq \theta\leq 1$, 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 Su\'arez-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 $2\leq \theta+2\beta$ 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.