Gevrey regularity for the Euler–Bernoulli beam equation with localized structural damping (Q6608464)

In this paper, the authors consider the Euler-Bernoulli beam equation \N\[\N\begin{cases} u_{tt}(x,t)+u_{xxxx}(x,t)-(a(x)u_{tx}(x,t))_x=0\\\Nx\in(0,\ell_0)\cup(\ell_0,\ell),t>0 \end{cases}\N\]\Nwith the localized discontinuous structural damping \N\[\Na(x)=\begin{cases} 0&\text{if }x\in(0,\ell_0)\\\N1&\text{if }x\in(\ell_0,\ell) \end{cases}.\N\]\NUnder some convenient transmission conditions at the interface and some convenient initial data, the authors propose to analyze both the stability and the regularity properties of the associated contraction \(C_0\)-semigroup \((S(t))_{t\geq0}\). In particular, they prove that \((S(t))_{t\geq0}\) is of Gevrey class \(\delta>24\) for \(t>0\); hence, immediately differentiable, which implies an instantaneous smoothing effect on the initial data.\N\NMoreover, they show that \((S(t))_{t\geq0}\) is exponentially stable.