Orientadores: Lucas Catão de Freitas Ferreira, Marcelo da Silva Montenegro, José Antonio Carrillo de la Plata / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T15:13:15Z (GMT). No. of bitstreams: 1 Santos_MatheusCorreiados_D.pdf: 865476 bytes, checksum: 31a8b558231b701d81c20bf2712e4f50 (MD5) Previous issue date: 2015 / Resumo: Analisaremos dois problemas neste trabalho. Na primeira parte, estudaremos a existência de soluções para uma equação elíptica semilinear no espaço euclidiano todo e com dependência do gradiente e onde nenhuma restrição é imposta sobre o comportamento da não linearidade no infinito. Provaremos que existe uma solução que é localmente única e que herda muitas das propriedades de simetria da não linearidade. A positividade da solução e seu comportamento assintótico também são analisados. Os resultados obtidos também podem ser estendidos para outros casos como o de domínios exteriores ou o semiespaço e também para alguns operadores fracionários. Na segunda parte, analisaremos o comportamento assintótico das soluções da versão fracionária unidimensional da equações de meios porosos introduzida por Caffarelli e Vázquez e onde a pressão é obtida como a inversa do laplaciano fracionário da densidade. Devido à convexidade do núcleo do potencial de Riesz em dimensão um, mostraremos que a entropia associada à equação é displacement convex e satisfaz uma desigualdade funcional envolvendo a dissipação da entropia e a distância de transporte euclidiana. Um argumento por aproximação mostra que essa desigualdade funcional é suficiente para deduzir que a entropia das soluções converge exponencialmente para a entropia do estado estacionário. Também provaremos uma nova desigualdade de interpolação que permitirá obter a convergência exponencial das soluções em espaços Lp / Abstract: We analyse two problems in this work. In the first part we study the existence of solutions to a semilinear elliptic equation in the whole space and with dependence on the gradient and where no restriction is imposed on the behavior of the nonlinearity at infinity. We prove that there exists a solution which is locally unique and inherits many of the symmetry properties of the nonlinearity. Positivity and asymptotic behavior of the solution are also addressed. Our results can be extended to other domains like half-space and exterior domains and also to some fractional operators. For the second part, we analyse the asymptotic behavior of solutions to the one dimensional fractional version of the porous medium equation introduced by Caffarelli and Vázquez and where the pressure is obtained as the inverse of the fractional Laplacian of the density. Due to the convexity of the kernel of the Riesz potential in one dimension, we show that the entropy associated with the equation is displacement convex and satisfies a functional inequality involving also entropy dissipation and the Euclidean transport distance. An argument by approximation shows that this functional inequality is enough to deduce the exponential convergence, in the entropy level, of solutions to the unique steady state. A new interpolation inequality is also proved in order to obtain the exponential decay also in Lp spaces / Doutorado / Matematica / Doutor em Matemática

Cette thèse est consacrée à l’étude des propriétés dynamiques des solutions de type soliton d'équations aux dérivées partielles (EDP) dispersives non linéaires. `A travers des exemples-type de telles équations, l'équation de Schrödinger non-linéaire (NLS), l'équation de Korteweg-de Vries généralisée (gKdV) et le système de Schrödinger, on traite du comportement des solutions convergeant en temps grand vers des sommes de solitons (multi-solitons). Dans un premier temps, nous montrons que dans une configuration symétrique, avec des interactions fortes, le comportement de séparation des solitons logarithmique en temps est universel à la fois dans le cas sous-critique et sur-critique pour (NLS). Ensuite, en adaptant les techniques précédentes à l'équation (gKdV), nous prouvons un résultat similaire de l'existence de multi-solitons avec distance relative logarithmique; pour (gKdV), les solitons sont répulsifs dans le cas sous-critique et attractifs dans le cas sur-critique. Finalement, nous identifions un nouveau régime de distance logarithmique où les solitons sont non-symétriques pour le système de Schrödinger non-intégrable; une telle solution n'existe pas dans le cas intégrable pour le système et pour (NLS). / This thesis deals with long time dynamics of soliton solutions for nonlinear dispersive partial differential equation (PDE). Through typical examples of such equations, the nonlinear Schrödinger equation (NLS), the generalized Korteweg-de Vries equation (gKdV) and the coupled system of Schrödinger, we study the behavior of solutions, when time goes to infinity, towards sums of solitons (multi-solitons). First, we show that in the symmetric setting, with strong interactions, the behavior of logarithmic separation in time between solitons is universal in both subcritical and supercritical case. Next, adapting previous techniques to (gKdV) equation, we prove a similar result of existence of multi-solitons with logarithmic relative distance; for (gKdV), the solitons are repulsive in the subcritical case and attractive in the supercritical case. Finally, we identify a new logarithmic regime where the solitons are non-symmetric for the non-integrable coupled system of Schrödinger; such solution does not exist in the integrable case for the system and for (NLS).

在這篇論文中，我們將考慮2個具有初值及邊界值的非線性波方程。首先，考慮一個具有某種阻尼項 (強阻尼、 線性阻尼及非線性阻尼) 的積分--微分方程。我們利用 Fadeo-Galerkin及 Contraction Mapping Principle的方法來建立局部存在性和唯一性，並且使用 Nako 的不等式 ([40]) 來討論解的長時間存在 (global existence) 及漸進行為( asymptotic behavior) 。至於在解的有限時間爆增 (finite time blow-up) 方面，我們使用直接方法 ([33]) 來探討具有強阻尼及線性阻尼的問題。另一方面，我們利用能量法 (energy method) 來討論非線性阻尼問題的有限時間爆增現象。其次，我們考慮一個具有特殊邊界值的 Kirchhoff方程， 我們利用擾動的能量法 (perturbed energy method) ([56]) 來研究解的漸進行為，並且使用直接方法 ([33]) 來探討解的有限時間暴增問題。最後，我們提出一些與本文相關的有趣問題以作為未來的研究。 / In this thesis, we shall consider two initial-boundary value problems for nonlinear wave equations. First, we consider a nonlinear integro- differential equation with some kind of damping terms - the strong damping term or the linear damping term or the nonlinear damping term. We establish the existence and uniqueness of local solutions by using Faedo-Galerkin method and Contraction Mapping Principle. We shall discuss the asymptotic behavior of global solutions by using Nako’s inequality ([40]). Moreover, the blow-up properties of local solutions with non-positive initial energy and small positive initial energy for strong or linear damping case are obtained by using direct method ([33]). On the other hand, for the nonlinear damping case, we apply the energy method to deduce the blow-up of local solutions with negative initial energy, vanishing initial energy and small positive initial energy. The estimates of lifespan of solutions are also given in each case. Secondly, we shall consider an initial-boundary value problem for a wave equation of Kirchhoff type with a linear boundary damping term. The asymptotic behavior of global solutions is investigated by using perturbed energy method ([56]). Moreover, the blow-up phenomena with the initial energy being non-positive and positive and the estimates for the blow-up time are obtained by direct approach ([33]). Finally, a list of some interesting problems related to our model is posed for further research.

This thesis consists of a summary and five papers, dealing with financial applications of optimal stopping, optimal control and volatility. In Paper I, we present a method to recover a time-independent piecewise constant volatility from a finite set of perpetual American put option prices. In Paper II, we study the optimal liquidation problem under the assumption that the asset price follows a geometric Brownian motion with unknown drift, which takes one of two given values. The optimal strategy is to liquidate the first time the asset price falls below a monotonically increasing, continuous time-dependent boundary. In Paper III, we investigate the optimal liquidation problem under the assumption that the asset price follows a jump-diffusion with unknown intensity, which takes one of two given values. The best liquidation strategy is to sell the asset the first time the jump process falls below or goes above a monotone time-dependent boundary. Paper IV treats the optimal dividend problem in a model allowing for positive jumps of the underlying firm value. The optimal dividend strategy is of barrier type, i.e. to pay out all surplus above a certain level as dividends, and then pay nothing as long as the firm value is below this level. Finally, in Paper V it is shown that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential Lévy models is the existence of jumps towards the strike price in the underlying process.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-23T16:00:02Z No. of bitstreams: 1 arquivototal.pdf: 9090532 bytes, checksum: d4fefb1d97e9c6d585d5d18a33abf752 (MD5) / Made available in DSpace on 2017-08-23T16:00:02Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 9090532 bytes, checksum: d4fefb1d97e9c6d585d5d18a33abf752 (MD5) Previous issue date: 2016-11-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis we study controllability results, asymptotic behavior and inverse problem related to some problems of the theory of partial di erential equations. Two particular systems are the focus of the study: the Mindin-Timoshenko system, describing the vibrational motion of a plate or a beam, and the phase eld system describing the temperature and phase of a medium having two distinct physical states. The rst chapter is devoted to the study of the 1-D Mindlin-Timoshenko system with discontinuous coe cient. A Carleman inequality is obtained under the assumption of monotonicity on the beam speed. Subsequently, two applications are provided: the controllability of the control system acting on the boundary and Lipschitzian stability of the inverse problem of recovering a potential from a single measurement of the solution. In the second chapter we consider a contact problem characterized by the behavior of a two-dimensional plate whose board makes contact with a rigid obstacle. The formulation of this problem is presented by the 2-D Mindlin-Timoshenko system with boundary conditions and suitable damping terms. Concerning such system, is proved via penalty techniques, the existence of solution and that the system energy has exponential decay when the time approaches in nity. In the third chapter, the study is aimed at a nonlinear phase- eld system de ned in a real open interval. Here we present some controllability results when a single control acts, by means of Dirichlet conditions, on the temperature equation of the system on one of the endpoints of the interval. To prove the results is used the method of moments, plus a spectral study of operators associated to the system and xed point theory to deal with the nonlinearity. / Nesta tese estudamos resultados de controlabilidade, comportamento assintótico e problema inverso relacionados a alguns problemas da teoria de equações diferenciais parciais. Dois sistemas particulares são foco do estudo: o sistema de Mindin-Timoshenko, que descreve o movimento vibratório de uma placa ou viga, e o sistema de campo de fases que descreve a temperatura e a fase de um meio onde ocorrem dois estados físicos distintos. O primeiro capítulo é dedicado ao estudo do sistema de Mindlin-Timoshenko 1-D com coe ciente descontínuos. Uma desigualdade de Carleman é obtida sob a hipótese de monotonicidade sobre velocidade da viga. Posteriormente, são fornecidas duas aplicações: a controlabilidade do sistema com controles agindo na fronteira e a estabilidade Lipschitziana do problema inverso de recuperar um potencial através de uma única informação obtida sobre a solução. No segundo capítulo consideramos um problema de contato caracterizado pelo comportamento de uma placa bidimensional cujo bordo faz contato com um obstáculo rígido. A formulação deste problema é apresentada pelo sistema de Mindlin-Timoshenko 2-D com condi ções de fronteira e termos de amortecimento (damping) adequados. Sobre tal sistema, é provada, através de técnicas de penalização, a existência de solução e, posteriormente, que sua energia possui decaimento exponencial quando o tempo tende ao in nito. No terceiro capítulo o estudo é voltado a um sistema de campo de fases não-linear de nido em um intervalo aberto real. Neste espaço apresentamos alguns resultados de controlabilidade quando um único controle age, sob condições de Dirichlet, na equação da temperatura em um dos bordos do intervalo. Para provar os resultados é utilizado o método dos momentos, além de uma estudo espectral de operadores associados ao sistema e teoria de ponto xo para lidar com a não-linearidade.

Après une présentation des notations usuelles de la théorie des graphes, on étudie l'ensemble des fonctions harmoniques sur les graphes, c'est à dire des fonctions dont le laplacien est nul. Ces fonctions forment un espace vectoriel et sur un graphe uniformément localement fini, on montre que cet espace vectoriel est soit de dimension un, soit de dimension infinie. Lorsque le graphe comporte une infinité de cycles, ce résultat tombe en défaut et on exhibe des exemples qui montrent qu'il existe un graphe sur lequel les harmoniques forment un espace vectoriel de dimension n, pour tout n. Un exemple de graphe périodique est également traité. Ensuite, toujours pour le laplacien, on étudie plus précisément sur les arbres uniformément localement finis les valeurs propres dont l'espace propre est de dimension infini. Dans ce cas, il est montré que l'espace propre contient un sous-espace isomorphe à l'ensemble des suites réelles bornées. Une inégalité concernant le spectre est donnée dans le cas spécial où les arêtes sont de longueur un. Des exemples montrent que ces inclusions sont optimales. Dans le chapitre suivant, on étudie le comportement asymptotique des valeurs propres pour des opérateurs elliptiques d'ordre 2 quelconques sous des conditions de Kirchhoff dynamiques. Après réécriture du problème sous la forme d'un opérateur de Sturm-Liouville, on écrit le problème de façon matricielle. Puis on trouve une équation caractéristique dont les zéros correspondent aux valeurs propres. On en déduit une formule pour l'asymptotique des valeurs propres. Dans le dernier chapitre, on étudie la stabilité de solutions stationnaires pour certains problèmes de réaction-diffusion où le terme de non linéarité est polynomial. / After a quick presentation of usual notations for the graph theory, we study the set of harmonic functions on graphs, that is, the functions whose laplacian is zero. These functions form a vectorial space. On a uniformly locally finite tree, we shaw that this space has dimension one or infinity. When the graph has an infinite number of cycles, this result change and we describe some examples showing that there exists a graph on which the harmonic functions form a vectorial space of dimension n, for all n. We also treat the case of a particular periodic graph. Then, we study more precisely the eigenvalues of infinite dimension. In this case, the eigenspace contains a subspace isomorphic to the set of bounded sequences. An inequality concerning the spectral is given when edges length is equal to one. Examples show that these inclusions are optimal. We also study the asymptotic behavior of eigenvalues for elliptic operators under dynamical Kirchhoff node conditions. We write the problem as a Sturm-Liouville operator and we transform it in a matrix problem. Then we find a characteristic equation whose zeroes correspond to eigenvalues. We deduce a formula for the asymptotic behavior. In the last chapter, we study the stability of stationary solutions for some reaction-diffusion problem whose the non-linear term is polynomial.

La première partie de cette thèse (articles I et II) est consacrée à l'étude du comportement asymptotique des solutions d'un système dynamique du second ordre avec dissipation évanescente. Le système dynamique est étudié dans sa version continue et dans sa version discrète via un algorithme.La deuxième partie de cette thèse (articles III à VI) est consacrée à l'étude de plusieurs algorithmes de type proximal. Nous montrons que ces algorithmes convergent vers des solutions de certains problèmes de minimisation. Dans chaque cas, une application est donnée dans le cadre de la décomposition de domaine pour les EDP. / The first part of this thesis is devoted to the study of the asymptotic behavior of solutions of a second order dynamic system with vanishing dissipation. The dynamic system is studied in its continuous version and in its discrete version via an algorithm.The second part is about the study of several proximal-type algorithms. We show that these algorithms converge to solutions of some minimization problems. In each case, an application is given in the area of domain decomposition for PDE's.

Dans cette thèse nous étudions l'influence de l'environnement sur le comportement d'une cellule dans deux situations différentes. Dans chacune de ces deux situations, apparaît un couplage non-linéaire sur le champ d'advection lié à un terme non-local provenant du bord du domaine. Dans une première partie, nous modélisons la polarisation cellulaire durant la conjugaison de la cellule de levure. Nous utilisons un modèle de type convection-diffusion avec un terme de convection non-linéaire et non-local. Ce modèle présente des similarités avec le modèle de Keller-Segel, la source du potentiel attractif étant sur le bord du domaine. Nous étudions le cas de la dimension un en utilisant des inégalités de Sobolev logarithmiques et HWI. En nous appuyant sur un raisonnement heuristique, nous ramenons l'étude de notre modèle en dimension deux au bord du domaine. Nous validons le modèle à l'aide des résultats expérimentaux obtenus par M. Piel en utilisant un bruit dynamique dans nos simulations numériques. Nous étudions ensuite le problème du dialogue cellulaire entre cellules de levure de sexe opposé. Dans une seconde partie, nous étudions la réaction immunitaire durant l'athérosclérose. Nous construisons puis développons un modèle structuré en âge pour décrire l'inflammation. Pour des paramètres particuliers, nous déterminons le comportement en temps long de notre système en utilisant une fonctionnelle de Lyapunov. / We investigate the influence of the environment on the behaviour of a cell in two different situations. In each of these situations, there is a non-linear coupling of the drift due to a non-local term coming from the boundary of the domain.The first part focuses on the modeling of cell polarisation during the mating of yeast. We use a convection-diffusion model with a non-linear and non-local drift. This model is similar to the Keller-Segel model, the source of the attractive potential comes from the boundary of the domain. We study the long time behaviour of the one-dimensional case by using logarithmic Sobolev and HWI inequalities.By relying on a heuristic, we reduce the study of our model in the two-dimensional case to the boundary of the domain. We validate the model with data provided by M. Piel. This validation requires adding a dynamical noise in our numerical simulations. We study then the cell discussion between yeast of opposite gender. In the second part we study the immune response in atherosclerosis. We build and then develop an age structured model in order to describe the inflammation. For specific parameters, we investigate the long time behaviour of our system by using a Lyapunov functional.

Dans cette thèse, nous étudions l'approximation et la stabilisation de certaines équations d'évolution, en utilisant la théorie des semi-groups et l'analyse spectrale. Cette thèse est divisée en deux parties principales. Dans la première partie, comme dans [3, 4], nous considérons l'approximation des équations d'évolution du deuxième ordre modélisant les vibrations de structures élastiques. Il est bien connu que le système approché par éléments finis ou différences finies n'est pas uniformément exponentiellement ou polynomialement stable par rapport au paramètre de discrétisation, même si le système continu a cette propriété. Dans la première partie, notre objectif est d'amortir les modes parasites à haute fréquence en introduisant des termes de viscosité numérique dans le schéma d'approximation. Avec ces termes de viscosité, nous montrons la décroissance exponentielle ou polynomiale du schéma discret lorsque le problème continu a une telle décroissance et quand le spectre de l'opérateur spatial associé au problème conservatif satisfait la condition du gap généralisée. En utilisant le Théorème de Trotter-Kato, nous montrons la convergence de la solution discrète vers la solution continue. Quelques exemples sont également présentés. / In this thesis, we study the approximation and stabilization of some evolution equations, using semigroup theory and some spectral analysis. This Ph.D. thesis is divided into two main parts. In the first part, as in [3, 4], we consider the approximation of second order evolution equations modeling the vibrations of elastic structures. It is well known that the approximated system by finite elements or finite differences is not uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Therefore, our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay and when the spectrum of the spatial operator associated with the undamped problem satisfies the generalized gap condition. By using the Trotter-Kato Theorem, we further show the convergence of the discrete solution to the continuous one. Some illustrative examples are also presented.

Disertační práce formuluje asymptotické odhady řešení tzv. sublineárních a superlineárních diferenciálních rovnic se zpožděním. V těchto odhadech vystupuje řešení pomocných funkcionálních rovnic a nerovností. Dále práce pojednává o kvalitativních vlastnostech diferenčních rovnic se zpožděním, které vznikly diskretizací studovaných diferenciálních rovnic. Pozornost je věnována souvislostem asympotického chování řešení rovnic ve spojitém a diskrétním tvaru, a to v obecném i v konkrétních případech. Studována je rovněž stabilita numerické diskretizace vycházející z $\theta$-metody. Práce obsahuje několik příkladů ilustrujících dosažené výsledky.