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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$.

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