Holomorphic solution semigroups for Sobolev-type equations in locally convex spaces

The problem of the existence of an exponentially bounded solution semigroup strongly holomorphic in a sector is studied for a Sobolev-type linear equation
\begin{equation} L\dot u=Mu \end{equation}
with continuous operator $L\colon\mathfrak U\to\mathfrak F$, $\ker L\ne\{0\}$, and closed densely defined operator $M\colon\operatorname{dom}M\to\mathfrak F$, where $\mathfrak U$ and $\mathfrak F$ are sequentially complete locally convex spaces. It is shown that the condition of the $(L,p)$-sectoriality of the operator $M$, which generalizes the well known condition of sectoriality, is necessary and sufficient for the existence of such semigroups degenerate at the $M$-associated vectors of the operator $L$ of height $p$ and lower and the existence of pairs of invariant subspaces of the operators $L$ and $M$. Generalizations of Yosida's theorem and results on the existence of a holomorphic solution semigroup for equation (1) in Banach spaces are obtained. These results are used in the study of the weakened Cauchy problem for equation (1) and for the corresponding non-linear equation. One application of the abstract results is a theorem on sufficient conditions for the solubility of the Cauchy problem for a class of equations in Fréchet spaces of a special kind. It is used in the analysis of the periodic Cauchy problem for a partial differential equation with displacement not solved with respect to the time derivative.