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Degenerate strongly continuous operator groups. (English. Russian original) Zbl 0998.47024
Russ. Math. 44, No. 3, 51-62 (2000); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2000, No. 3, 54-65 (2000).
The degenerate Cauchy problem for the linear differential-operator equation $Lu'(t) =M u(t),$ $$\ker L\not=\{0\}$$ on the whole $$\mathbb{R}$$ is considered. Here $$L\in {\mathcal L}(U,F)$$ is bounded linear operator and $$M\in {\mathcal Cl}(U,F)$$ is closed linear one in Banach spaces $$U, F$$. It is proved that if $$M$$ is $$(L,p)$$-biradial: \begin{aligned}\max\{\|R_{(\lambda,p)}^n\|, \|L_{(\lambda,p)}^n\|\}\leq \frac K{{\prod}_{k=0}^{p}(|\lambda_k|-a)^n}, \quad \lambda=(\lambda_1,...\lambda_p), \lambda_k\in \mathbb{R},|\lambda_k|>a, \lambda_k\in\rho^L(M),\end{aligned}
\begin{aligned} R_{(\lambda,p)}:={\prod}_{k=0}^{p}(\lambda_k L-M)^{-1}L, \quad L_{(\lambda,p)}:={\prod}_{k=0}^{p}L(\lambda_k L-M)^{-1}),\end{aligned} then for the Cauchy problem there exists an exponentially bounded strongly continuous solving group on a subspace $$\tilde{U}$$ of $$U$$. Under some additional estimates the decomposition of $$\tilde{U}=U$$ is obtained. The results extends well known results for the equation $$u'(t) =A u(t)$$ and continue previous ones of author.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 34G10 Linear differential equations in abstract spaces