In this paper we establish the regularity, exponential stability of global (weak) solutions and existence of uniform compact attractors of semiprocesses, which are generated by the global solutions, of a two-parameter family of operators for the nonlinear 1-d non-autonomous viscoelasticity. We employ the properties of the analytic semigroup to show the compactness for the semiprocess generated by the global solutions.

The thesis consists of two parts, a theoretical part and an applied part, and in addition an Appendix. Except for a very short chapter in the applied part and the appendix we only present previously unknown results leading to a very concise style. In the theoretical part we study rates of decay for vector-valued functions and semigroups of operators depending on a real and positive variable. Under boundedness assumptions on the function/semigroup itself and under analytic extendability assumptions of its Laplace transform/resolvent across the imaginary axis we provide (almost) sharp rates of decay. Our results improve known results in this very active area of research. In the second part of the thesis we apply our results to specific examples (from the field of PDEs): local energy decay for wave equations on exterior domains, energy decay for damped wave equations on bounded domains and decay for a viscoelastic boundary damping model for sound waves. Many more examples can be found in the vast literature.

In der Dissertation werden Werkzeuge zur Analyse von Wohlgestelltheit und Asymptotik von Integro-Differential- und Verzögerungsgleichungen entwickelt. Im ersten Teil der Arbeit (Kapitel 1 und 2) werden Methoden zur Bestimmung der Modulhalbgruppe (kleinste dominierende C0-Halbgruppe) einer C0-Halbgruppe zur Verfügung gestellt, die unter anderem auf Volterra-Halbgruppen (die aus Integro-Differentialgleichungen hervorgehen) und Evolutionshalbgruppen (Rückkopplungsgleichungen mit Zeitverzögerung, Transport in Netzwerken) angewendet werden. Im Mittelpunkt des zweiten Teils (Kapitel 3 und 4) steht ein Integro-Differentialgleichungstyp, der Schwingungsphänomene von Tragswerksflächen im Unterschallbereich beschreibt. Das besondere dieser Gleichung ist das Auftreten der Zeitableitung der gesuchten Funktion im Integralterm. Es werden eine Reihe von Wohlgestelltheitskriterien hergeleitet, welche Wohlgestelltheit der Gleichung liefern, ohne das es möglich ist, durch partielle Integration die Zeitableitung im Integralterm zu beseitigen und dadurch die Gleichung auf einen bekannten Integro-Differentialgleichungstyp zurückzuführen. Die entwickelten Methoden eignen sich auch für die Herleitung neuer Wohlgestelltheitskriterien für andere Verzögerungsgleichungen. Entsprechende Resultate werden in Kapitel 4 hergeleitet. / In the dissertation tools for the analysis of well-posedness and asymptotic behaviour of integro-differential equations and delay equations are developed. In the first part (chapter 1 and 2) methods for the determination of the modulus semigroup (smallest dominating C0-semigroup) of a C0-semigroup are provided and applied to various examples such as Volterra semigroups and evolution semigroups and transport evolution equations in networks. The main interest of the second part (chapter 3 and 4) is a type of an integro-differential equation which occurs in the modelling of the flutter of airfoils at subsonic speed. The remarkable property of the equation is the time derivative of the sought function in the integral term. A number of well-posedness criteria are proved for which integration by parts is not possible. The developed methods are also suitable for the derivation of new well-posedness results for other delay semigroups. Corresponding criteria are presented in chapter 4.

The thesis consists of two parts, a theoretical part and an applied part, and in addition an Appendix. Except for a very short chapter in the applied part and the appendix we only present previously unknown results leading to a very concise style. In the theoretical part we study rates of decay for vector-valued functions and semigroups of operators depending on a real and positive variable. Under boundedness assumptions on the function/semigroup itself and under analytic extendability assumptions of its Laplace transform/resolvent across the imaginary axis we provide (almost) sharp rates of decay. Our results improve known results in this very active area of research. In the second part of the thesis we apply our results to specific examples (from the field of PDEs): local energy decay for wave equations on exterior domains, energy decay for damped wave equations on bounded domains and decay for a viscoelastic boundary damping model for sound waves. Many more examples can be found in the vast literature.:Part 1 Quantified Tauberian theorems and decay of C0-semigroups 1 Decay of vector-valued functions 2 Optimal decay for C0-semigroups on Hilbert spaces Part 2 Applications: decay of waves 3 Local decay for waves in exterior domains 4 Waves on a square with constant damping on a strip 5 A viscoelastic boundary damping model

In der Dissertation werden Werkzeuge zur Analyse von Wohlgestelltheit und Asymptotik von Integro-Differential- und Verzögerungsgleichungen entwickelt. Im ersten Teil der Arbeit (Kapitel 1 und 2) werden Methoden zur Bestimmung der Modulhalbgruppe (kleinste dominierende C0-Halbgruppe) einer C0-Halbgruppe zur Verfügung gestellt, die unter anderem auf Volterra-Halbgruppen (die aus Integro-Differentialgleichungen hervorgehen) und Evolutionshalbgruppen (Rückkopplungsgleichungen mit Zeitverzögerung, Transport in Netzwerken) angewendet werden. Im Mittelpunkt des zweiten Teils (Kapitel 3 und 4) steht ein Integro-Differentialgleichungstyp, der Schwingungsphänomene von Tragswerksflächen im Unterschallbereich beschreibt. Das besondere dieser Gleichung ist das Auftreten der Zeitableitung der gesuchten Funktion im Integralterm. Es werden eine Reihe von Wohlgestelltheitskriterien hergeleitet, welche Wohlgestelltheit der Gleichung liefern, ohne das es möglich ist, durch partielle Integration die Zeitableitung im Integralterm zu beseitigen und dadurch die Gleichung auf einen bekannten Integro-Differentialgleichungstyp zurückzuführen. Die entwickelten Methoden eignen sich auch für die Herleitung neuer Wohlgestelltheitskriterien für andere Verzögerungsgleichungen. Entsprechende Resultate werden in Kapitel 4 hergeleitet. / In the dissertation tools for the analysis of well-posedness and asymptotic behaviour of integro-differential equations and delay equations are developed. In the first part (chapter 1 and 2) methods for the determination of the modulus semigroup (smallest dominating C0-semigroup) of a C0-semigroup are provided and applied to various examples such as Volterra semigroups and evolution semigroups and transport evolution equations in networks. The main interest of the second part (chapter 3 and 4) is a type of an integro-differential equation which occurs in the modelling of the flutter of airfoils at subsonic speed. The remarkable property of the equation is the time derivative of the sought function in the integral term. A number of well-posedness criteria are proved for which integration by parts is not possible. The developed methods are also suitable for the derivation of new well-posedness results for other delay semigroups. Corresponding criteria are presented in chapter 4.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we first develop a brief theoretical approach of semigroups of bounded linear operators, culminating on Hille-Yosida Theorem. Then we used the extrapolation theory to study su cient conditions to obtain existence and uniqueness of Almost Automorphic and Pseudo-Almost Automorphic mild solutions, through the Banach's Fixed Point Theorem for the semilinear evolution equation x(t) = Ax(t) + f(t; x(t)); t E R, where A : D(A) X ! X is a Hille-Yosida operator of negative type and not necessary dense domain on the Banach space X. / Neste trabalho, desenvolvemos inicialmente uma breve abordagem te orica dos semigrupos de operadores lineares limitados, culminando no Teorema de Hille-Yosida. Em seguida, usamos a teoria de extrapolação a fim de estudar condições suficientes para obtermos a existência e a unicidade de soluções brandas Quase Automórficas e Pseudo-quase Automórficas, por meio do Teorema do Ponto Fixo de Banach, para a equação de evolução semilinear x(t) = Ax(t) + f(t; x(t)); t E R, onde A : D(A) X ! X é um operador de Hille-Yosida de tipo negativo e dom ínio não necessariamente denso, definido no espaço de Banach X.

<p>Doktorska disertacija je posvećena primeni teorije polugrupa operatora na re&scaron;avanje dve klase Cauchy-jevih početnih problema. U prvom delu smo<br />ispitivali parabolične stohastičke parcijalne diferencijalne jednačine (SPDJ-ne), odredjene sa dva tipa operatora: linearnim zatvorenim operatorom koji<br />generi&scaron;e <em>C</em>₀&minus;polugrupu i linearnim ograničenim operatorom kombinovanim<br />sa Wick-ovim proizvodom. Svi stohastički procesi su dati Wiener-It&ocirc;-ovom<br />haos ekspanzijom. Dokazali smo postojanje i jedinstvenost re&scaron;enja ove klase<br />SPDJ-na. Posebno, posmatrali smo i stacionarni slučaj kada je izvod po<br />vremenu jednak nuli. U drugom delu smo konstruisali kompleksne stepene<br /><em>C</em>-sektorijalnih operatora na sekvencijalno kompletnim lokalno konveksnim<br />prostorima. Kompleksne stepene operatora smo posmatrali kao integralne<br />generatore uniformno ograničenih analitičkih <em>C</em>-regularizovanih rezolventnih<br />familija, i upotrebili dobijene rezultate na izučavanje nepotpunih Cauchy-jevih problema vi&scaron;3eg ili necelog reda.</p> / <p>The doctoral dissertation is devoted to applications of the theory<br />of semigroups of operators on two classes of Cauchy problems. In the first<br />part, we studied parabolic stochastic partial differential equations (SPDEs),<br />driven by two types of operators: one linear closed operator generating a<br /><em>C</em>₀&minus;semigroup and one linear bounded operator with Wick-type multipli-cation. All stochastic processes are considered in the setting of Wiener-It&ocirc;<br />chaos expansions. We proved existence and uniqueness of solutions for this<br />class of SPDEs. In particular, we also treated the stationary case when the<br />time-derivative is equal to zero. In the second part, we constructed com-plex powers of <em>C</em>&minus;sectorial operators in the setting of sequentially complete<br />locally convex spaces. We considered these complex powers as the integral<br />generators of equicontinuous analytic <em>C</em>&minus;regularized resolvent families, and<br />incorporated the obtained results in the study of incomplete higher or frac-tional order Cauchy problems.</p>