*(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

where $\xb7$ means $\frac{d}{dt}$, $y,{y}_{0},{y}_{1}\in H$, a Hilbert space with inner product ($\xb7,\xb7)$ and associated with $\parallel \xb7\parallel $, A and B are positive definite self-adjoint unbounded linear operators on H and B is ${A}^{1/2}$-bounded. Letting ${x}_{1}={A}^{1/2}y$, ${x}_{2}=\dot{y}$, we get the equivalent first order linear systems

G. Chen and D. L. Russel (*) have proved that $\mathcal{A}=\left(\begin{array}{cc}0& {A}^{1/2}\\ -{A}^{1/2}& -B\end{array}\right)$ 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 $\mathcal{D}\left(B\right)\supset \mathcal{D}\left({A}^{1/2}\right)$, such that

or

then $\mathcal{A}=\left(\begin{array}{cc}0& {A}^{1/2}\\ -{A}^{1/2}& -B\end{array}\right)$ 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 |