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On the holomorphic property of the semigroup associated with linear elastic systems with structural damping. (English) Zbl 0636.73047

G. Chen and D. L. Russell [(*) A mathematical model for linear elastic systems with structural damping, MRC Technical summary Report 2089, Math. Res. Center. Univ. Wisconsin-Madison (1980)] studied the following linear elastic systems with structural damping

$\left\{\begin{array}{c}\stackrel{¨}{y}+B\stackrel{˙}{y}+Ay=0,\hfill \\ y\left(0\right)={y}_{0},\stackrel{˙}{y}\left(0\right)={y}_{1},\hfill \end{array}\right\$

where $·$ means $\frac{d}{dt}$, $y,{y}_{0},{y}_{1}\in H$, a Hilbert space with inner product ($·,·\right)$ and associated with $\parallel ·\parallel$, A and B are positive definite self-adjoint unbounded linear operators on H and B is ${A}^{1/2}$-bounded. Letting ${x}_{1}={A}^{1/2}y$, ${x}_{2}=\stackrel{˙}{y}$, we get the equivalent first order linear systems

$\left\{\begin{array}{c}\frac{d}{dt}\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)=\left(\begin{array}{cc}0& {A}^{1/2}\\ -{A}^{1/2}& -B\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)=𝒜\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)\hfill \\ {x}_{1}\left(0\right)={A}^{1/2}{y}_{0},\phantom{\rule{2.em}{0ex}}{x}_{2}\left(0\right)={y}_{1}·\hfill \end{array}\right\$

G. Chen and D. L. Russel (*) have proved that $𝒜=\left(\begin{array}{cc}0& {A}^{1/2}\\ -{A}^{1/2}& -B\end{array}\right)$ generates a holomorphic semigroup, if some addition conditions are satisfied. Moreover, they still conjectured that if A and B are positive definite self-adjoint unbounded linear operators with the domain $𝒟\left(B\right)\supset 𝒟\left({A}^{1/2}\right)$, such that

$\left(I\right)\phantom{\rule{1.em}{0ex}}{\beta }_{1}\left({A}^{1/2}x,x\right)\le \left(Bx,x\right)\le {\beta }_{2}\left({A}^{1/2}x,x\right),\phantom{\rule{1.em}{0ex}}x\in 𝒟\left({A}^{1/2}\right),$

or

$\left(II\right)\phantom{\rule{1.em}{0ex}}{\beta }_{1}\left(Ax,x\right)\le \left(B\phantom{\rule{1.em}{0ex}}2x,x\right)\le {\beta }_{2}\left(Ax,x\right),\phantom{\rule{1.em}{0ex}}x\in 𝒟\left(A\right),$

then $𝒜=\left(\begin{array}{cc}0& {A}^{1/2}\\ -{A}^{1/2}& -B\end{array}\right)$ should generate a holomorphic semi-group, where ${\beta }_{1}$ and ${\beta }_{2}$ are positive constant numbers with ${\beta }_{1}\le {\beta }_{2}$. Also some partial results for conjecture (I) and (II) are shown in (*). Recently, the author gave an answer to the conjecture (I) affirmatively [A problem for linear elastic systems with structural damping (to appear)]. In this paper we will show that conjecture (II) is also true.

##### MSC:
 74H45 Vibrations (dynamical problems in solid mechanics) 47D03 (Semi)groups of linear operators 47B25 Symmetric and selfadjoint operators (unbounded) 20M05 Free semigroups, generators and relations, word problems