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

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 \[ {\mathcal {A}_B} = \left | {\begin {array}{*{20}{c}} 0 & I \\ { - A} & { - B} \\ \end {array} } \right |\] (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 $t > 0$.