We study a Dirichlet problem, investigating existence and uniqueness for semipositone and superlinear nonlinearities. We make use of Pohozaev identities, energy arguments, and bifurcation from a simple eigenvalue.

Nous étudions des équations relatives au p-Laplacien anisotrope lorsque certaines composantes du vecteur p sont égales à 1. / We study anisotropic p-Laplacian equations when some components of p are equal to 1.

Dans cette thèse, on s'intéresse au problème de Cauchy pour des équations quasi-linéaires dispersives. Pour une telle équation, l'enjeu est de montrer l'existence et l'unicité d'une solution de l'équation avec une donnée initiale prescrite dans un espace fonctionnel le plus large possible. Nous étudierons deux modèles décrivant l'évolution de la surface d'un fluide satisfaisant certaines conditions physiques.La première partie est consacrée à l'étude de l'équation de Kadomtsev-Petviashvili avec forte tension de surface (KP-I). Cette équation possède une structure Hamiltonienne et admet donc une fonctionnelle d'énergie préservée par le flot. Afin d'obtenir des solutions définies globalement en temps, on cherche donc à construire un flot dans l'espace de Banach naturellement associé à cette énergie. De plus, on se restreint à des espaces contenant des solutions particulières (les solitons linéaires de KdV), on impose donc une condition de périodicité dans la direction transverse à la propagation du fluide.On commence par illustrer le caractère quasi-linéaire de l'équation en montrant a priori que le flot dans cet espace ne peut pas être très régulier. Ceci restreint l'éventail des méthodes connues pour résoudre ce type de problème. On a donc recours à la méthode dite de restriction de la transformée de Fourier en temps petits développée récemment par Ionescu, Kenig et Tataru pour traiter ce même modèle sans condition de périodicité. On obtient ainsi l'existence globale et l'unicité de la solution du problème de Cauchy dans l'espace d'énergie. Enfin, on montre que le flot ainsi construit est continu mais pas uniformément continu sur les ensembles bornés de l'espace d'énergie.Une application intéressante de la construction d'un flot global sur l'espace d'énergie contenant les solitons linéaires est de lever une restriction sur les perturbations admissibles dans un résultat de Rousset-Tzvetkov sur la stabilité orbitale des solitons linéaires de faible vitesse.Dans la deuxième partie de la thèse, on s'intéresse à l'équation KP-I d'ordre cinq, qui est une alternative au modèle précédent dans le cas d'une tension de surface avoisinant une valeur critique pour laquelle l'effet dispersif devient plus faible. Pour cette équation, le comportement quasi-linéaire ne se manifeste que pour des données périodiques dans la direction transverse, et les autres cas avaient été étudiés précédemment dans les travaux de Saut et Tzvetkov. On considère ici des données également périodiques dans la direction de propagation. On montre que pour certains choix de périodes, le flot ne peut pas être régulier. Afin de traiter le problème indifféremment des périodes spatiales, on utilise donc une nouvelle fois la méthode précédente pour construire un flot global dans l'espace associé au Hamiltonien de ce modèle. / This thesis investigates the Cauchy problem for some quasilinear dispersive equations. Being given such an equation, the goal is then to construct a unique solution to this equation with a prescribed initial data belonging in a function space as large as possible. We will study two models describing the time evolution of the surface of a fluid in a particular regime.The first part of this thesis is devoted to the study of the Kadomtsev-Petviashvili equation in the case of strong surface tension (KP-I). This equation has a Hamiltonian structure, so it admits an energy functional which is preserved under the flow. In order to recover solutions which are globally defined in time, we thus seek to construct a flow map in the Banach sace naturally associated with the energy. In addition, we restrict ourself to spaces including some special solutions (the KdV line soliton), so we require the functions to be periodic in the transverse direction.We start by illustrating the quasilinear behaviour of the equation : we show that a flow map defined on this space cannot be too regular. This limits the range of applicable methods known to solve this kind of problem. We thus use the so-called small times Fourier restriction norm method recently developped by Ionescu, Kenig and Tataru to deal with the same model without the periodicity assumption. We thereby obtain the global existence and uniqueness of a solution to the Cauchy problem in the energy space. At last, we prove that the flow map constructed this way is continuous yet not uniformly continuous on the bounded sets of the energy space.An interesting application of the construction of a global flow on the energy space containing the line solitons is to get rid of an extra condition on admissible perturbations in a result of Rousset-Tzvetkov on the orbital stability of the small speed line solitons.In the second part of the thesis, we turn to the fifth-order KP-I equation, which is an alternative to the previous model should the tension surface come close to a critical value in which the dispersive effect becomes weaker. Regarding this equation, the quasilinear behaviour only manifests when solutions are periodic in the transverse direction, and the other cases were treated in the work of Saut and Tzvetkov. We study the case of functions which are also periodic in the direction of propagation, and we show that at least for some choice of periods the flow map fails to be smooth. In order to treat the problem regardless of the periods, we make another use of the method above to construct a global flow in the space associated to the Hamiltonian of the equation.

In this paper, the monotone method is extended to the initial-boundary value problems of nonlocal PDE system of first order, both quasi-monotone and non-monotone. A comparison principle is established, and a monotone scheme is given.

En aquesta tesi es té en compte un model no lineal de creixement de població de cèl·lules que s'estructuren pel seu contingut de ciclina i cinases depenents de ciclina (CDK). Aquest model condueix a un sistema no lineal d'equacions en derivades parcials de primer ordre amb termes no locals. Per estudiar aquest sistema utilitzem la teoria de semigrups lineals positius i la formulació semilineal, que són eines molt poderoses per fer front a l'anàlisi d'aquest tipus de models, tant des del punt de vista del problema de valor inicial, com de l'existència i l'estabilitat d'estats estacionaris. El model que es considera a la tesi descriu la següent situació biològica: les cèl·lules s'estructuren en relació amb el contingut d'un determinat grup de proteïnes anomenades ciclines i CDK i es divideixen en dos tipus: proliferants i quiescents. Les cèl·lules proliferants creixen i es divideixen, donant a lloc al final del cicle cel·lular a noves cèl·lules, o bé van cap al compartiment de les quiescents, mentre que les cèl·lules quiescents no envelleixen ni es divideixen, ni canvien el seu contingut de ciclina, però o tornen cap al compartiment de proliferació o bé romanen en l’estat de repòs. D'altra banda, tant les cèl·lules proliferants com les quiescents poden experimentar l'apoptosi, la mort cel·lular programada. L'únic terme no lineal en el model és un terme de reclutament de cèl·lules quiescents cap a la fase de proliferació. En aquest treball demostrem l'existència global, unicitat i positivitat de les solucions del problema de valor inicial. Reescrivint el nostre sistema en una forma abstracta podem demostrar que un cert operador lineal és el generador infinitesimal d'un semigrup positiu fortament continu. D'altra banda s'utilitza la formulació semilineal estàndard per a l’equació no lineal abstracta i obtenim una única solució global positiva per a qualsevol condició inicial positiva a L1. També es prova l'existència i unicitat d'un estat estacionari no trivial del nostre sistema sota hipòtesis adequades. Com es fa sovint en situacions similars, el problema és relacionat amb provar l'existència (i unicitat) d'un vector propi positiu normalitzat. Això correspon als vectors propis del valor propi dominant d'un determinat operador lineal positiu parametritzat pel valor de la variable de feedback. L'existència tant del valor propi dominant i de (l’únic) vector propi positiu està donat per una versió del teorema de Perron- Frobenius en dimensió infinita. També s’inclouen simulacions numèriques basades en la integració al llarg de les línies característiques. Amb l'ajuda d'aquestes simulacions numèriques trobem inestabilitat de l'estat estacionari per a valors de paràmetres compatibles amb els que donen inestabilitat en el model de dimensió finita. També s'inclou la demostració de l'existència de solucions independents del contingut de ciclina per a una elecció molt particular dels valors dels paràmetres i funcions que defineixen el model. Finalment s'utilitza la formulació anomenada cumulativa (o en retard) de la dinàmica de poblacións estructurades. En particular s'ha considerat una versió diferent del model estudiat abans, on es suposa que el pas de proliferants a quiescents només pot ocórrer una sola vegada, enfocament oposat al primer model on aquestes transicions poden ocórrer infinites vegades. A més a més, també suposem que hi ha un valor particular x del contingut de ciclina que separa les cèl·lules que encara no es poden dividir de les altres que sí que poden dividir-se. L'equació del model resulta ser una equació amb retard que relaciona els valors actuals d'aquestes variables amb la seva història (el seu valor en el passat). Fent servir aquest enfocament, es pot provar l'existència i unicitat de solucions del problema de valor inicial, i el principi d'estabilitat lineal a través d'una formulació semilineal en el marc dels semigrups duals. / In this thesis we consider a nonlinear cell population model where cells are structured with respect to the content of cyclin and cyclin dependent kinases (CDK). This model leads to a first order nonlinear partial differential equations system with non local terms. To study this system we use the theory of positive linear semigroups and the semilinear formulation, which are very powerful tools to deal with the analysis of this kind of models, both from the point of view of the initial value problem as well as the existence and stability of steady states. The model considered in the thesis describes the following biological situation: cells are structured with respect to the content of a certain group of proteins called cyclin and CDK and are distributed into two types: proliferating and quiescent cells. The proliferating cells grow and divide, giving birth at the end of the cell cycle to new cells, or else transit to the quiescent compartment, whereas quiescent cells do not age nor divide nor change their cyclin content but either transit back to the proliferating compartment or else stay in the quiescent compartment. Moreover, both proliferating and quiescent cells may experiment apoptosis, i.e. programmed cell death. The only nonlinear term is a recruitment term of quiescent cells going back to the proliferating phase. In this work we start proving global existence, uniqueness and positiveness of the solutions of the initial value problem. We rewrite our system in an abstract form and show that some linear operator is the infinitesimal generator of a positive strongly continuous semigroup. On the other hand we use the standard semilinear formulation for the nonlinear (abstract) equation and obtain a unique global positive solution for any positive initial condition in L1. We also prove the existence and uniqueness of a nontrivial steady state of our system under suitable hypotheses. As it is often done in similar situations, the problem is related to proving the existence (and uniqueness) of a positive normalized eigenvector. This eigenvector corresponds to the dominant eigenvalue of a certain positive linear operator parameterized by the value of the (one dimensional) feedback variable G. The existence of both dominant eigenvalue and (unique) positive eigenvector is given by a version of the infinite dimensional Perron-Frobenius theorem. We include numerical simulations based on the integration along characteristic lines. With the help of these numerical simulations we find instability of the steady state for parameter values compatible with the ones which give instability in the finite dimensional model. We also include a computation showing the existence of cyclin-independent solutions for a very particular choice of the parameter values and functions defining the model. Finally we use the so-called cumulative or delayed formulation of the structured population dynamics. In particular we have considered a different version of the model studied before, where one assumes that proliferating cells can become quiescent only once opposed to the other approach where these transitions can occur infinitely many times and moreover, we also assume that there is a particular value x of the cyclin content that separates cells which still cannot divide from the others which are able to divide. The model equation turns out to be a delay equation relating the current values of these variables with their history (their value in the past). Using this approach, one can prove existence and uniqueness of solutions of the initial value problem, and the linear stability principle by means of a semi-linear formulation in the framework of dual semigroups.

In this paper, we examine a method on how to model and produce Chladni Figures. We walk through how a thin metal plate, when vibrating at certain frequencies, can create various interest patterns. First we discuss the equation for the vertical force exerted on the plate, then we derive a PDE to solve for the nodal lines (lines that remain fixed, while the rest of the plate is oscillating). And, discuss how to create and model these figures, through a finite difference method. There have been several experiments on Chladni Figures, using some sort of vibrating membrane or plate and then either through the use of a speaker or a violin bow, produce frequencies in order to resonate with the membrane. These eigenvalue solutions can been physically observed by putting sand on the plate and vibrating it. We will approximate theses figures, calculate the convergence of the approximation, and relate the generated figures to figures produced in experiments.

Aspects of the asymptotic behaviour of cell-growth models described by partial differential equations, and systems of partial differential equations, are considered. The models considered describe the evolution of the size-distribution or age-distribution of a population of cells undergoing growth and division. First, the relationship between the behaviour, with and without dispersion, of a single-compartment size-distribution model of cell-growth with fixed-size cell division (where cells can only divide at a single, critical size) is considered. In this model dispersion accounts for stochastic variation in the growth process of each individual cell. Existence, uniqueness and the asymptotic stability of the solution is shown for a size-distribution model of cell-growth with dispersion and fixed-size cell division. The conditions for the analysis to hold for a more general class of division behaviours are also discussed. A class of nonlocal ordinary differential equations is studied, which contains as a subset the nonlocal ordinary differential equations describing the steady size-distributions of a single-compartment model of cell-growth. Existence of solutions to these equations is found to be implied by the existence of 'upper' and 'lower' solutions, which also provide bounds for the solution. A multi-compartment, age-distribution model of cell-growth is studied, which describes the evolution of the age-distribution of cells in different phases of cell-growth. The stability of the model when periodic solutions exist is examined. Sufficient conditions are given for the existence of stable steady age-distributions, as well as for stable periodic solutions. Finally, a multi-compartment age-size distribution model of cell-growth is studied, which describes the evolution of the age-size distribution of cells in different phases of cell-growth. Sufficient conditions are given for the existence of steady age-size distributions. An outline of the analysis required to prove stability of the steady age-size distributions of the model is also given. The analysis is based on ideas introduced in the previous chapters.

No description available.

O PDE Escola, objeto desta pesquisa, é um programa do Ministério da Educação, de apoio à gestão escolar baseado no planejamento participativo e destinado a auxiliar as escolas públicas a melhorar sua gestão, conforme informado em seus materiais. O enfoque de pesquisa foi na edição 2011, em que o programa priorizou as escolas cujo IDEB 2009 foi igual ou inferior à média nacional e que não tinham sido priorizadas entre 2008 e 2010. Em linhas gerais, esta pesquisa teve como objetivo compreender qual a representação de problemas da escola para o desenvolvimento da educação, por sua perspectiva, considerando o nível de distanciamento ou aproximação existente em relação à proposta da política pública educacional do PDE Escola. Para isso, serviram como fontes documentais, as publicações oficiais do Ministério da Educação, a partir de 2006, quando a política começou a se expandir envolvendo as escolas públicas com os IDEBs mais baixos em todo o país, até 2011, que é o período do grupo de escolas priorizadas no PDE Escola, selecionado para análise. As fontes ou referências principais para análise foram os Planos de Desenvolvimento de Escolas da Rede Estadual do Rio Grande do Sul, priorizadas em 2011, que faziam parte da última edição do programa, ainda disponível para acesso em sistema on line do MEC em 2013. Como amostra para esta pesquisa foi definido o grupo das 44 escolas que tiveram o PDE Escola aprovado em 2011, com execução do recurso financeiro em 2012. A abordagem do ciclo de políticas de Stephen J. Ball foi a perspectiva de análise utilizada, mas com enfoque no contexto da prática. Fizeram parte do aporte teórico as abordagens sobre: ideologia; sociologia da globalização; a educação e o estado capitalista; o novo gerencialismo; o modelo de gestão do bem-estar social e o novo gerencialismo; as políticas e gestão da educação básica e escolar no Brasil. Por fim, verificou-se por meio deste estudo, que as dimensões do diagnóstico do PDE Escola, das escolas da amostra com as médias percentuais maiores na indicação de problemas e problemas considerados críticos, foram as dos indicadores e taxas, e taxas de rendimento, seguidas das que tratam da infraestrutura e comunidade escolar, sendo que ao final, com as médias percentuais mais baixas, ficaram as de ensino e aprendizagem e por último a de gestão. Isso evidenciou a forte influência da proposta da política sobre o que as escolas consideram como problemas na educação, que repercute até hoje nas dimensões em que se concentram as médias percentuais de escolas mais elevadas, na indicação de problemas. A proposta da política pesquisada neste estudo demonstrou enfocar a necessidade de avanço sobre os resultados educacionais, por meio da responsabilização dos atores locais para o planejamento de ações que solucionem isso, mas a representação de problemas apresentados aponta contradições que revelam outras necessidades, apesar de até certo ponto a proposta da política se materializar na priorização de problemas pela maioria das escolas, nas dimensões que são o foco do MEC. / The PDE School, object of this research is a program of the Ministry of Education to support school management based on participatory planning and designed to assist the public schools to improve their management, as reported in their materials. The research focus was in the 2011 edition, in which the program prioritized schools whose IDEB 2009 was equal to or below the national average and that had not been prioritized between 2008 and 2010. In general, this research aimed to understand what the representation school problems for the development of education, in perspective, considering the level of detachment or existing approach to the proposal of public education policy of PDE school. For that, served as documentary sources, the official publications of the Ministry of Education, from 2006, when the policy began to expand involving public schools with the lowest IDEBs across the country by 2011, which is the period of group prioritized schools in PDE School, selected for analysis. The sources or major references for analysis were the Development Plans of Schools of State Network of Rio Grande do Sul, prioritized in 2011, which were part of the last edition of the program, still available for access online system MEC in 2013. As sample for this study was defined the group of 44 schools that had the PDE School approved in 2011, with implementation of financial resources in 2012. the approach of Stephen J. Ball policies cycle analysis perspective was used, but with a focus on context of practice. Were part of the theoretical framework approaches on: ideology; sociology of globalization; education and the capitalist state; the new managerialism; the social welfare management model and the new managerialism; policies and management in primary and school education in Brazil. Finally, it has been found by this study that the dimensions of the diagnosis of PDE School Schools of the sample with the highest percentage means the indication considered critical issues and problems, and the indicators were the rates and yields, followed by dealing with infrastructure and school community, and at the end, with the lowest percentage averages, were the teaching and learning and finally the management. This showed the strong influence of the proposed policy on what schools consider as problems in education that affects today in dimensions that concentrate the percentage means higher schools, the indication of problems. The proposed policy researched in this study demonstrated focus on the need to advance on educational outcomes, through the empowerment of local actors for planning actions that solve this, but the representation of the problems presented points contradictions that reveal other needs, although so some extent the policy proposal to materialize in prioritizing problems by most schools, dimensions that are the focus of MEC.

In this essay we present the Sobolev spaces and some basic properties of them. The Sobolev spaces serve as a theoretical framework for studying solutions to partial differential equations. The finite element method is presented which is a numerical method for solving partial differential equations.