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arXiv · 2310.00869

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

Santos R. W. S. Bejarano·Filomena B. R. Mendes·Fredy M. Sobrado Suárez·Gilson Tumelero·Marieli M. Tumelero
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Abstract

In this work we study a strongly coupled system between the equation of plates with fractional rotational inertial force $κ(-Δ)^βu_{tt}$ where the parameter $0 <β\leq 1$ and the equation of an electrical network containing a fractional dissipation term $δ(-Δ)^θv_t$ where the parameter $0\leq θ\leq 1$, the strong coupling terms are given by the Laplacian of the displacement speed $γΔu_t$ and the Laplacian electric potential field $γΔv_t$. When $β= 1$, we have the Kirchoff-Love plate and when $β= 0$, 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 $(θ,β)\in [0,1]\times(0,1]-\{( 1,1/2)\}$. We also determine two Gevrey classes: $s_1 >\frac{1}{2\max\{ \frac{1-β}{3-β}, \fracθ{2+θ-β}\}}$ for $2\leq θ+2β$ and $s_2> \frac{2(2+θ-β)}θ$ when the parameters $θ$ and $β$ lies in the interval $(0, 1)$ and we finish by proving that at the point $(θ,β)=(1,1/2)$ 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.

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Santos R. W. S. Bejarano, Filomena B. R. Mendes, Fredy M. Sobrado Suárez, Gilson Tumelero, Marieli M. Tumelero. 2023-10-02. The regularity of the coupled system between an electrical network with fractional dissipation and a plate equation with fractional inertial rotational. https://arxiv.org/abs/2310.00869

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