Integrated semigroups.

*(English)*Zbl 0604.47025
Mathematische Fakultät der Eberhard-Karls-Universität zu Tübingen. 120 p. (1986).

The paper under review is based on the concept of so-called ”integrated semigroups” introduced recently by W. Arendt. A closed (not necessarily densly defined) operator A is called the generator of the integrated semigroup \((S_ t)_{t\geq 0}\) (i.e. a strongly continuous family of bounded linear operators satisfying some axioms) if \((\lambda -A)^{- 1}/\lambda\) is the Laplace transform of \((S_ t)_ t.\)

The main point is that if A is such a generator then the abstract Cauchy problem (ACP) \[ (i)\quad \dot u(t)=Au(t)\quad (t\geq 0),\quad (ii)\quad u(0)=u_ 0\in {\mathcal D}(A) \] possesses a unique solution, explicitely given by \(u(t)=\frac{d}{dt}S_ t(u_ 0).\)

In this thesis the author develops a perturbation theory for such operators A and - more important - applies the abstracts theory to very concrete problems involving partial differential operators.

Compared with the theory of distribution semigroups (initialized by J. Lions) the present theory seems to be more easily applicable. Thus Kellermann’s results on the solvability of the above mentioned ACP for certain differential operators are not known within the frame of distribution semigroups.

The main point is that if A is such a generator then the abstract Cauchy problem (ACP) \[ (i)\quad \dot u(t)=Au(t)\quad (t\geq 0),\quad (ii)\quad u(0)=u_ 0\in {\mathcal D}(A) \] possesses a unique solution, explicitely given by \(u(t)=\frac{d}{dt}S_ t(u_ 0).\)

In this thesis the author develops a perturbation theory for such operators A and - more important - applies the abstracts theory to very concrete problems involving partial differential operators.

Compared with the theory of distribution semigroups (initialized by J. Lions) the present theory seems to be more easily applicable. Thus Kellermann’s results on the solvability of the above mentioned ACP for certain differential operators are not known within the frame of distribution semigroups.

Reviewer: M.Wolff

##### MSC:

47D03 | Groups and semigroups of linear operators |

47B44 | Linear accretive operators, dissipative operators, etc. |

47F05 | General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |