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Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $ 0<\alpha <\frac 12$


Authors: Shu Ping Chen and Roberto Triggiani
Journal: Proc. Amer. Math. Soc. 110 (1990), 401-415
MSC: Primary 47D05; Secondary 34G10, 73D30
DOI: https://doi.org/10.1090/S0002-9939-1990-1021208-4
MathSciNet review: 1021208
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Abstract: Let $ A$ (the elastic operator) be a positive, self-adjoint operator with domain $ \mathcal{D}(A)$ in the Hilbert space $ X$, and let $ B$ (the dissipation operator) be another positive, self-adjoint operator satisfying $ {\rho _1}{A^\alpha } \leq B \leq {\rho _2}{A^\alpha }$ for some constants $ 0 < {\rho _1} < {\rho _2} < \infty $ and $ 0 < \alpha \leq 1$. Consider the operator

$\displaystyle {\mathcal{A}_B} = \left\vert {\begin{array}{*{20}{c}} 0 & I \\ { - A} & { - B} \\ \end{array} } \right\vert$

(corresponding to the elastic model $ \ddot x + B\dot x + Ax = 0$ written as a first order system), which (once closed) is plainly the generator of a strongly continuous semigroup of contractions on the space $ E = \mathcal{D}({A^{1/2}}) \times X$. In [C-T.l] [C-T.3] we showed that, for $ \tfrac{1}{2} \leq \alpha \leq 1$, such a semigroup is analytic (holomorphic) on $ E$ on a triangular sector of $ {\mathbf{C}}$ containing the positive real axis and, moreover, that the property of analyticity is false for $ 0 < \alpha < \tfrac{1}{2}$, say for $ B = {A^\alpha }$. We now complete the description of $ {\mathcal{A}_B}$ in the range $ 0 < \alpha < \tfrac{1}{2}$ by showing that such semigroup is in fact of Gevrey class $ \delta > 1/2\alpha $, hence differentiable on $ E$ for all $ t > 0$.

References [Enhancements On Off] (What's this?)

  • [B.1 ] A. V. Balakrishnan, Damping operators in continuum models of flexible structures, preprint.
  • [C-R.1] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39 (1982), 433-454. MR 644099 (83f:70034)
  • [C-T.1] S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: the case $ \alpha = \tfrac{1}{2}$, in Springer-Verlag Lecture Notes in Math., vol. 1354, pp. 234-256 (Proceedings of the Seminar in Approximation and Optimization, University of Habana, Cuba, January 10-12, 1987). MR 996678 (90j:34088)
  • [C-T.2] -, Proof of extensions of two conjectures on structural damping for elastic systems: the case $ \tfrac{1}{2} \leq \alpha \leq 1$, Pacific J. Math. 39 (1989), 15-55. MR 971932 (90g:47071)
  • [C-T.3] S. Chen and R. Triggiani, Analyticity of elastic systems with non selfadjoint dissipation: the general case, preprint, 1989. MR 971932 (90g:47071)
  • [K.1] S. G. Krein, Linear differential equations in Banach space, Trans. Amer. Math. Soc. 29 (1971). MR 0342804 (49:7548)
  • [K.2] T. Kato, Perturbation theory for linear operators, Springer-Verlag, 1966. MR 0203473 (34:3324)
  • [L.1] W. Littman, A generalization of a theorem of Datko and Pazy, University of Minnesota, preprint, 1988; presented at International Conference, Baton Rouge, Louisiana, October 19-21, 1988. MR 1029070 (91a:47052)
  • [L.2] J. L. Lions, Controlabilite exacte des systemes distribues, Masson, Paris, 1989.
  • [L-M.1] J. L. Lions and E. Magenes, Non homogeneous boundary value problems and applications I, Springer-Verlag, New York, 1972.
  • [P.1] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
  • [T.l] R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl. 52 (1975), 383-403. Addendum J. Math. Anal. Appl. 56 (1976), 492-493. MR 0445388 (56:3730)
  • [T.2] -, Improving stability properties of hyperbolic damped equations by boundary feedback, Springer-Verlag Lecture Notes (LNCIS), 1985, 400-409.
  • [T.3] -, Finite rank, relatively bounded perturbations of semi-group generators, Part III: a sharp result on the lack of uniform stabilization, Differential and Integral Equations 3 (1990), 503-522. MR 1047750 (91f:93091)
  • [T.4] S. Taylor, Ph.D. thesis, Chapter 'Gevrey semigroups', School of Mathematics, University of Minnesota, 1989.

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DOI: https://doi.org/10.1090/S0002-9939-1990-1021208-4
Article copyright: © Copyright 1990 American Mathematical Society

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