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Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-22T13:00:30Z No. of bitstreams: 1 arquivototal.pdf: 1160851 bytes, checksum: 7135123aaf3823254b3e52ab8d141d6d (MD5) / Made available in DSpace on 2016-03-22T13:00:30Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1160851 bytes, checksum: 7135123aaf3823254b3e52ab8d141d6d (MD5) Previous issue date: 2015-03-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we present an introduction to the theory of C0-semigroup (and C0-group) of bounded linear operators, and we show that wave operator in RN is the in nitesimal generator of a C0-group of bounded linear operators in a appropriate Banach space. / Neste trabalho apresentamos uma introdu c~ao a teoria de C0􀀀semigrupos (e C0􀀀gru po) de operadores lineares e limitados, e mostramos que operador de ondas em RN e o gerador in nitesimal de um C0􀀀grupo de operadores lineares e limitados em um espa co de Banach apropriado.

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.

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.

Th.--Sci. math.--Lille 1, 1977. N°: 381.

In the Thesis, linear stochastic differential equations in a Hilbert space driven by a cylindrical fractional Brownian motion with the Hurst parameter in the interval H < 1/2 are considered. Under the conditions on the range of the diffusion coefficient, existence of the mild solution is proved together with measurability and continuity. Existence of a limiting distribution is shown for exponentially stable semigroups. The theory is modified for the case of analytical semigroups. In this case, the conditions for the diffusion coefficient are weakened. The scope of the theory is illustrated on the Heath-Jarrow-Morton model, the wave equation, and the heat equation. 1

Cette thèse est consacrée à l’étude de la stabilisation directe et indirecte de différents systèmes d’équations d’ondes avec un contrôle frontière de type fractionnaire ou un contrôle local viscoélastique de type Kelvin-Voight. Nous considérons, d’abord, la stabilisation de l’équation d’ondes multidimensionnel avec un contrôle frontière fractionnaire au sens de Caputo. Sous des conditions géométriques optimales, nous établissons un taux de décroissance polynomial de l’énergie de système. Ensuite, nous nous intéressons à l’étude de la stabilisation d’un système de deux équations d’ondes couplées via les termes de vitesses, dont une seulement est amortie avec contrôle frontière de type fractionnaire au sens de Caputo. Nous montrons différents résultats de stabilités dans le cas 1-d et N-d. Finalement, nous étudions la stabilité d’un système de deux équations d’ondes couplées avec un seul amortissement viscoélastique localement distribué de type Kelvin-Voight. / This thesis is devoted to study the stabilization of the system of waves equations with one boundary fractional damping acting on apart of the boundary of the domain and the stabilization of a system of waves equations with locally viscoelastic damping of Kelvin-Voight type. First, we study the stability of the multidimensional wave equation with boundary fractional damping acting on a part of the boundary of the domain. Second, we study the stability of the system of coupled onedimensional wave equation with one fractional damping acting on a part of the boundary of the domain. Next, we study the stability of the system of coupled multi-dimensional wave equation with one fractional damping acting on a part of the boundary of the domain. Finally, we study the stability of the multidimensional waves equations with locally viscoelastic damping of Kelvin-Voight is applied for one equation around the boundary of the domain.