Local \(C\)-cosine families and \(n\)-times integrated local cosine families
For the investigation of ill-posed first-order abstract Cauchy problems, the theory of \(C\)-semigroups and \(n\)-times integrated semigroups has been developed and intensively studied in the last 20 years. The main technical tool in these investigations was the exhausting use of vector-valued Laplace transforms theory. Recently, an analogous line of investigations has been started to investigate second-order (in time) abstract Cauchy problems. The paper under review continues these investigations and develops a theory for local \(C\)-cosine families and \(n\)-times integrated local cosine families. Here, the word ``local'' refers to the fact that the cosine family may be only defined on a finite time interval. Generation theorems for the complete infinitesimal generator in analogy with the Hille-Yosida theory for \(C_0\)-semigroups are proven.
scientific article; zbMATH DE number 2148914