Primene polugrupa operatora u nekim klasama Košijevih početnih problema

Abstract

Doktorska disertacija je posvećena primeni teorije polugrupa operatora na rešavanje dve klase Cauchy-jevih početnih problema. U prvom delu smo ispitivali parabolične stohastičke parcijalne diferencijalne jednačine (SPDJ-ne), odredjene sa dva tipa operatora: linearnim zatvorenim operatorom koji generiše C0−polugrupu i linearnim ograničenim operatorom kombinovanim sa Wick-ovim proizvodom. Svi stohastički procesi su dati Wiener-Itô-ovom haos ekspanzijom. Dokazali smo postojanje i jedinstvenost rešenja ove klase SPDJ-na. Posebno, posmatrali smo i stacionarni slučaj kada je izvod po vremenu jednak nuli. U drugom delu smo konstruisali kompleksne stepene C-sektorijalnih operatora na sekvencijalno kompletnim lokalno konveksnim prostorima. Kompleksne stepene operatora smo posmatrali kao integralne generatore uniformno ograničenih analitičkih C-regularizovanih rezolventnih familija, i upotrebili dobijene rezultate na izučavanje nepotpunih Cauchy-jevih problema viš3eg ili necelog reda.

The doctoral dissertation is devoted to applications of the theory of semigroups of operators on two classes of Cauchy problems. In the first part, we studied parabolic stochastic partial differential equations (SPDEs), driven by two types of operators: one linear closed operator generating a C0−semigroup and one linear bounded operator with Wick-type multipli-cation. All stochastic processes are considered in the setting of Wiener-Itô chaos expansions. We proved existence and uniqueness of solutions for this class of SPDEs. In particular, we also treated the stationary case when the time-derivative is equal to zero. In the second part, we constructed com-plex powers of C−sectorial operators in the setting of sequentially complete locally convex spaces. We considered these complex powers as the integral generators of equicontinuous analytic C−regularized resolvent families, and incorporated the obtained results in the study of incomplete higher or frac-tional order Cauchy problems.