Abstract degenerate Cauchy problems in locally convex spaces (Q5945934)

The authors consider the following abstract degenerate Cauchy problem NEWLINE\[NEWLINE{d\over dt} Bu(t)= Au(t),\quad t\geq 0,\quad Bu(0)= Bu_0.NEWLINE\]NEWLINE Here, \(A\), \(B\) are closed operators in a sequentially complete locally convex space topologized by a family of seminorms. For the problem, they introduce a \(C\)-propagation family \(\{W(t)\}_{t\geq 0}\) that is a generalization of a \(C\)-regularized semigroup, so, of a strongly continuous semigroup as well.NEWLINENEWLINENEWLINEIn terms of \(\{W(t)\}_{t\geq 0}\), they derive a general \(C\)-wellposedness result regarding the problem.NEWLINENEWLINENEWLINENext, they obtain conditions ensuring the existence of \(C\)-propagation families for those problems with differential operators, on various function spaces, as coefficient operators. Even for Banach spaces the results are new.