Coercive differential operators and fractionally integrated cosine functions. (Q1607717)

The author studies the integrated semigroups based on fractional integration. Let \(X\) be a Banach space, \(B(X)\) the space of linear bounded operators in \(X\) and \(\rho(B)\) denote the resolvent set of \(B\). An operator \(B\in B(X)\) is said to generate an \(\alpha\)-times integrated cosine function \(C(t)\), \(t\geq 0\), if there exists \(\alpha \geq 0\) such that for large \(\lambda\in \mathbb{R}\), \(\lambda^2 \in \rho(B)\) and \(\lambda{1-\alpha(\lambda^2 I -B)^{-1}}\) is the Laplace transform of \(C(t)\), where \(C:[0,\infty)\to B(X)\) is assumed to be an exponentially bounded and strongly continuous family. The main statement of the paper is the following result. Let a polynomial \(P(\xi)\) be \(r\)-coercive (which means that \(| P(\xi)| ^{-1}=O(| \xi| ^{-r})\) as \(| \xi| \to\infty\)) for some \(r\in(0,m)\) and let \(\omega=\sup_{\xi \in \mathbb{R}^n}P(\xi)< \infty\) . If \(\overline{\rho(P(A))}\neq 0\), then \(\overline P(A)\) generates a norm-continuous \(\alpha\)-times integrated cosine function \(C(t)\), in case \(\alpha> n\frac{2m-r}{2r}\). Besides this, \(\| C(t)\| \leq M \left(1+t^\frac{n}{2}\right)e^{\sqrt\omega t}\) when \(\omega>0\) with some power estimate depending on \(\alpha\) when \(\omega \leq 0\).