[pt] Consideramos a análise numérica de perturbações não lineares do Laplaciano definido em regiões limitadas tratáveis pelo Método de Elementos Finitos. Supomos que as não linearidades interagem com k autovalores do Laplaciano livre. Apresentamos uma redução do problema à inversão de uma função de k variáveis e delineamos uma técnica para tal. O texto é uma extensão dos trabalhos de Cal Neto, Malta, Saldanha e Tomei. / [en] We consider the numerical analysis of nonlinear perturbations of the Laplacian defined in limited regions amenable to the Finite Element Method. The nonlinearities are supposed to interact only with k eigenvalues of the free Laplacian. We present a reduction of the problem to the inversion of a function of k variables and indicate a technique to do so. The text extends the works by Cal Neto, Malta, Saldanha and Tomei.

This thesis is concerned with the derivation of similarity solutions for one-dimensional coupled systems of reaction - diffusion equations, a semi-linear system and a one-dimensional tripled system. The first area of research in this thesis involves a coupled system of diffusion equations for the existence of two distinct families of diffusion paths. Constructing one-parameter transformation groups preserving the invariance of this system of equations enables similarity solutions for this coupled system to be derived via the classical and non-classical procedures. This system of equation is the uncoupled in the hope of recovering further similarity solutions for the system. Once again, one-parameter groups leaving the uncoupled system invariant are obtained, enabling similarity solutions for the system to be elicited. A one-dimensional pattern formation in a model of burning forms the next component of this thesis. The primary focus of this area is the determination of similarity solutions for this reaction - diffusion system by means of one-parameter transformation group methods. Consequently, similarity solutions which are a generalisation of the solutions of the one-dimensional steady equations derived by Forbes are deduced. Attention in this thesis is then directed toward a semi-linear coupled system representing a predator - prey relationship. Two approaches to solving this system are made using the classical procedure, leading to one-parameter transformation groups which are instrumental in elicting the general similarity solution for this system. A triple system of equations representing a one-dimensional case of diffusion in the presence of three diffusion paths constitutes the next theme of this thesis. In association with the classical and non-classical procedures, the derivation of one-parameter transformation groups leaving this system invariant enables similarity solutions for this system to be deduced. The final strand of this thesis involves a one- dimensional case of the general linear system of coupled diffusion equations with cross-effects for which one-parameter transformation group methods are once more employed. The one-parameter groups constructed for this system prove instrumental in enabling the attainment of similarity solutions for this system to be accomplished / Faculty of Business and Technology

The main goal of our thesis is to understand qualitative properties of solutions to the Cauchy problem for the semi-linear wave model with time-dependent speed and dissipation. We greatly benefited from very precise estimates for the corresponding linear problem in order to obtain the global existence (in time) of small data solutions. This reason motivated us to introduce very carefully a complete description for classification of our models: scattering, non-effective, effective, over-damping. We have considered those separately.

The dissertation consists of two parts. The first part (Chapter 1 to 4) is on some contributions to the development of a non-linear analysis on the quintessential fractal set Sierpinski gasket and its probabilistic interpretation. The second part (Chapter 5) is on the asymptotic tail decays for suprema of stochastic processes satisfying certain conditional increment controls. Chapters 1, 2 and 3 are devoted to the establishment of a theory of backward problems for non-linear stochastic differential equations on the gasket, and to derive a probabilistic representation to some parabolic type partial differential equations on the gasket. In Chapter 2, using the theory of Markov processes, we derive the existence and uniqueness of solutions to backward stochastic differential equations driven by Brownian motion on the Sierpinski gasket, for which the major technical difficulty is the exponential integrability of quadratic processes of martingale additive functionals. A Feynman-Kac type representation is obtained as an application. In Chapter 3, we study the stochastic optimal control problems for which the system uncertainties come from Brownian motion on the gasket, and derive a stochastic maximum principle. It turns out that the necessary condition for optimal control problems on the gasket consists of two equations, in contrast to the classical result on ℝ^d, where the necessary condition is given by a single equation. The materials in Chapter 2 are based on a joint work with Zhongmin Qian (referenced in Chapter 2). Chapter 4 is devoted to the analytic study of some parabolic PDEs on the gasket. Using a new type of Sobolev inequality which involves singular measures developed in Section 4.2, we establish the existence and uniqueness of solutions to these PDEs, and derive the space-time regularity for solutions. As an interesting application of the results in Chapter 4 and the probabilistic representation developed in Chapter 2, we further study Burgers equations on the gasket, to which the space-time regularity for solutions is deduced. The materials in Chapter 4 are based on a joint work with Zhongmin Qian (referenced in Chapter 4). In Chapter 5, we consider a class of continuous stochastic processes which satisfy the conditional increment control condition. Typical examples include continuous martingales, fractional Brownian motions, and diffusions governed by SDEs. For such processes, we establish a Doob type maximal inequality. Under additional assumptions on the tail decays of their marginal distributions, we derive an estimate for the tail decay of the suprema (Theorem 5.3.2), which states that the suprema decays in a manner similar to the margins of the processes. In Section 5.4, as an application of Theorem 5.3.2, we derive the existence of strong solutions to a class of SDEs. The materials in this chapter is based on the work [44] by the author (Section 5.2 and Section 5.3) and an ongoing joint project with Guangyu Xi (Section 5.4).

Submitted by Livia Mello (liviacmello@yahoo.com.br) on 2016-09-23T20:38:51Z No. of bitstreams: 1 TeseWNN.pdf: 1247691 bytes, checksum: 63f743255181169a9bb4ca1dfd2312c2 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:35:27Z (GMT) No. of bitstreams: 1 TeseWNN.pdf: 1247691 bytes, checksum: 63f743255181169a9bb4ca1dfd2312c2 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:35:33Z (GMT) No. of bitstreams: 1 TeseWNN.pdf: 1247691 bytes, checksum: 63f743255181169a9bb4ca1dfd2312c2 (MD5) / Made available in DSpace on 2016-09-26T20:35:40Z (GMT). No. of bitstreams: 1 TeseWNN.pdf: 1247691 bytes, checksum: 63f743255181169a9bb4ca1dfd2312c2 (MD5) Previous issue date: 2016-02-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / In this thesis we study the asymptotic properties for the solution of the Cauchy problem for the Klein-Gordon equation with non-effective time-dependent potential. The main goal was define a suitable energy related to the Cauchy problem and derive decay estimates for such energy. Strichartz’ estimates and results of scattering and modified scattering was established. The C m theory and the stabilization condition was applied to treat the case where the coefficient of the potential term has very fast oscillations. Moreover, we consider a semi-linear wave model scale-invariant time- dependent with mass and dissipation, in this step we used linear estimates related with the semi-linear model to prove global existence (in time) of energy solutions for small data and we show a blow-up result for a suitable choice of the coefficients. / Nesta tese estudamos as propriedades assintóticas para a solução do problema de Cauchy para a equação de Klein-Gordon com potencial não efetivo dependente do tempo. O principal objetivo foi definir uma energia adequada relacionada ao problema de Cauchy e derivar estimativas para tal energia. Estimativas de Strichartz e resultados de scatering e scatering modificados também foram estabelecidos. A teoria C m e a condição de estabilização foram aplicados para tratar o caso em que o coeficiente da massa oscila muito rápido. Além disso, consideramos um mod- elo de onda semi-linear scale-invariante com massa e dissipação dependentes do tempo, nesta etapa usamos as estimativas lineares de tal modelo para provar ex- istência global (no tempo) de solução de energia para dados iniciais suficientemente pequenos e demonstramos um resultado de blow-up para uma escolha adequada dos coeficientes.

On étudie l'existence de blow-up et l'ensemble des points de blow-up pour une équation de type chaleur dégénérée ou non avec un terme source uniforme fonction nonlinéaire de la température instantanée en un point fixé du domaine. L'étude est conduite par les méthodes d'analyse classique (fonctions de Green, développements en fonctions propres, principe du maximum) ou fonctionnelle (semi-groupes d'opérateurs linéaires). / We study existence of blow-up and blow-up sets for a (degenerate or not) heat-like equation with a uniform source term non linear function of the instantaneous temperature at a given point of the domain. The techniques are relevant from either classical analysis (Green functions, eigenfunction expansions, maximum principle) or functional analysis (semi-groups of linear operators).

This thesis represents a theoretical and numerical investigation of the canonical duality theory, which has been recently proposed as an alternative to the classic and direct methods for non-convex variational problems. These non-convex variational problems arise in a wide range of scientific and engineering applications, such as phase transitions, post-buckling of large deformed beam models, nonlinear field theory, and superconductivity. The numerical discretization of these non-convex variational problems leads to global minimization problems in a finite dimensional space. The primary goal of this thesis is to apply the newly developed canonical duality theory to two non-convex variational problems: a modified version of Ericksen's bar and a problem of Landau-Ginzburg type. The canonical duality theory is investigated numerically and compared with classic methods of numerical nature. Both advantages and shortcomings of the canonical duality theory are discussed. A major component of this critical numerical investigation is a careful sensitivity study of the various approaches with respect to changes in parameters, boundary conditions and initial conditions. / Ph. D.

The main goal of our thesis is to understand qualitative properties of solutions to the Cauchy problem for the semi-linear wave model with time-dependent speed and dissipation. We greatly benefited from very precise estimates for the corresponding linear problem in order to obtain the global existence (in time) of small data solutions. This reason motivated us to introduce very carefully a complete description for classification of our models: scattering, non-effective, effective, over-damping. We have considered those separately.

Cette thèse aborde deux sujets de recherches, le premier est sur l’existence et l’unicité des solutions des Équations Différentielles Doublement Stochastiques Rétrogrades (EDDSRs) et les Équations aux Dérivées partielles Stochastiques (EDPSs) multidimensionnelles à croissance surlinéaire. Le deuxième établit l’existence d’un contrôle optimal strict pour un système controlé dirigé par des équations différentielles stochastiques progressives rétrogrades (EDSPRs) couplées dans deux cas de diffusions dégénérée et non dégénérée.• Existence et unicité des solutions des EDDSRs multidimensionnels :Nous considérons EDDSR avec un générateur de croissance surlinéaire et une donnée terminale de carré intégrable. Nous introduisons une nouvelle condition locale sur le générateur et nous montrons qu’elle assure l’existence, l’unicité et la stabilité des solutions. Même si notre intérêt porte sur le cas multidimensionnel, notre résultat est également nouveau en dimension un. Comme application, nous établissons l’existence et l’unicité des solutions des EDPS semi-linéaires.• Contrôle des EDSPR couplées :Nous étudions un problème de contrôle avec une fonctionnelle coût non linéaire dont le système contrôlé est dirigé par une EDSPR couplée. L’objective de ce travail est d’établir l’existence d’un contrôle optimal dans la classe des contrôle stricts, donc on montre que ce contrôle vérifie notre équation et qu’il minimise la fonctionnelle coût. La méthode consiste à approcher notre système par une suite de systèmes réguliers et on montre la convergence. En passant à la limite, sous des hypothèses de convexité, on obtient l’existence d’un contrôle optimal strict. on suit cette méthode théorique pour deux cas différents de diffusions dégénérée et non dégénérée. / In this Phd thesis, we considers two parts. The first one establish the existence and the uniquness of the solutions of multidimensional backward doubly stochastic differential equations (BDSDEs in short) and the stochastic partial differential equations (SPDEs in short) in the superlinear growth generators. In the second part, we study the stochastic controls problems driven by a coupled Forward-Backward stochastic differentialequations (FBSDEs in short).• BDSDEs and SPDEs with a superlinear growth generators :We deal with multidimensional BDSDE with a superlinear growth generator and a square integrable terminal datum. We introduce new local conditions on the generator then we show that they ensure the existence and uniqueness as well as the stability of solutions. Our work go beyond the previous results on the subject. Although we are focused on multidimensional case, the uniqueness result we establish is new in one dimensional too. As application, we establish the existence and uniqueness of probabilistic solutions tosome semilinear SPDEs with superlinear growth generator. By probabilistic solution, we mean a solution which is representable throughout a BDSDEs.• Controlled coupled FBSDEs :We establish the existence of an optimal control for a system driven by a coupled FBDSE. The cost functional is defined as the initial value of the backward component of the solution. We construct a sequence of approximating controlled systems, for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we get the existence of a feedback optimal control. The convexity condition is used to ensure that the optimal control is strict. In this part, we study two cases of diffusions : degenerate and non-degenerate.

Made available in DSpace on 2014-06-12T18:28:06Z (GMT). No. of bitstreams: 2 arquivo4092_1.pdf: 696485 bytes, checksum: 34f193833bda4c51476df064875faf7b (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2011 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Nesta tese estudamos o comportamento assintótico de equações funcionais com retardo, especificamente a existência de soluções brandas quase automórficas e quase periódicas para a equação diferencial parcial com retardo finito. Existência de soluções brandas assintoticamente quase automórficas para uma equação diferencial funcional semi-linear e uma quação integro-diferencial com retardo infinito também é tratada. E estudamos a existência de soluções S-assintóticamente ω-periódicas e assintoticamente quase automórficas para uma classe de equações integro-diferenciais abstratas, estabelecemos alguns resultados gerais sobre a existência de soluções brandas S-assintoticamente ω-periódicas para tais equações e damos aplicações de nossos resultados abstratos. Além disso, estabelecemos condições para a existência de soluções brandas S-assintoticamente ω-periódicas de uma equação integral especifica que aparece no estudo de condução do calor em materiais com memória e apresentamos resultados similares para equações parciais integro-diferenciais abstratas com retardo infinito; discutimos também a existência de soluções brandas S-assintoticamente ω-periódica e assintoticamente quase automórficas para a equação integro-diferencial abstrata neutra. Finalmente, como aplicação das idéias anteriores, o capítulo final é concernente com o problema de estabilização para sistemas de controle hereditário