## Coercive differential operators and fractionally integrated cosine functions.(English)Zbl 1037.47029

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 $$\omega \leq 0$$.

### MSC:

 47D09 Operator sine and cosine functions and higher-order Cauchy problems 47F05 General theory of partial differential operators
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