Strongly continuous semigroups for Sobolev type equations in locally convex spaces (Q2713892)

The author considers the semigroup corresponding to the Sobolev type equation \(L\dot{u}=Mu\), with \(L:U\to F\) a linear continuous operator and \(M:U\to F\) a linear closed densely defined operator. Here \(U\) and \(F\) are sequentially complete locally convex topological spaces. The operator \(L\) is not assumed to be invertible. The results of the article are generalizations of earlier author's results and the results by G.~A. Sviridyuk. The conditions ensuring the existence of semigroups and their strong continuity are stated in terms of behavior of the operator-functions \(\mu\to (\mu L-M)^{-1}L\) and \(\lambda\to L(\lambda L-M)^{-1}\). The results are applied to the study of initial-boundary value problems for the equation NEWLINE\[NEWLINE P_n(A)u_t(x,t)=Q_m(A)u(x,t),\quad (x,t)\in \Omega\times {\mathbb R}^+, NEWLINE\]NEWLINE where \(P_n\) and \(Q_m\) are polynomials of degrees \(n\) and \(m\) in \(A\) and \(A\) is a selfadjoint second order elliptic operator.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].