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

TL;DR

This paper investigates the regularity and analyticity of the semigroup generated by a coupled system involving fractional inertial and dissipation terms in a plate and electrical network, revealing conditions for different Gevrey classes and analyticity.

Contribution

It establishes the non-analyticity of the semigroup for most parameter values, identifies specific Gevrey classes, and proves analyticity at a critical parameter point, advancing understanding of fractional coupled systems.

Findings

Semigroup is not analytic for most parameter pairs.

Two Gevrey classes are explicitly determined based on parameters.

Semigroup is analytic at the point (1, 1/2).

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.