在這一篇論文中我們討論的是下列這個非線性初值問題： u''(t)=u'(t)^q(c_1+c_2u(t)^p) u(0) = u_0; u'(0) = u_1: 我們關注於上述問題正解的一些性質。我們發現了一些爆破(Blow-up)現象，並獲得一些結果，有關爆破率(Blow-up rate)、爆破常數(Blow-up constant)以及爆破時間(Blow-up time)。 / In this paper we study the following initial value problem for the nonlinear equation, u''(t)=u'(t)^q(c_1+c_2u(t)^p) u(0) = u_0; u'(0) = u_1: We are interested in properties of positive solutions of the above problem.We have found blow-up phenomena and obtained some results on blowup rates, blow-up constants and life-spans.

O filme Blow-up desloca os seus espectadores para uma situação de aporia: nossos sentidos são capazes de apreender o real tal como ele é, ou, pelo contrário, falseiam a realidade? O nosso conhecimento do mundo possibilita somente uma percepção limitada e carregada de valores? Este relativismo acompanhou o debate acerca da epistemologia e as ciências ao longo do tempo e passou por um afunilamento agudo na contemporaneidade. Para lançar luz sobre o tema, Nietzsche é um autor privilegiado e incontornável no sentido de elaborar uma perspectiva a respeito do conhecimento histórico. Ao mesmo tempo, Blow-up é uma metáfora epistemológica da filosofia nietzscheana. Logo, essa relação entre ficção, história e filosofia vai muito além do mero espelhamento e incide diretamente na maneira que se compreende as condições metodológicas e teóricas de pesquisa do historiador e, por conseguinte, da escrita da história. / Blow-up is a film that brings its viewers into contact with a situation of aporia: are our faculties capable to grasp what is real as it really is, or, do they distort reality? Does our knowledge about the world only permit a limited perception charged with values? This relativism accompanied the debate about epistemology and the sciences over time and went through a severe process of change in contemporaneity. Therefore, Nietzsche is a privileged and an unavoidable author, in the sense of elaborating a perspective on historical knowledge, to cast light on the subject. At the same time, Blow-up is an epistemology metaphor of the nietzschean philosophy. Consequently, the relationship among fiction, history and philosophy goes beyond reflecting because it affects directly on how research methodological and theoretical circumstances of the historians are understood, as well as how history is written.

Considere o problema de valor inicial e de fronteira \'u IND.t\'= \'delta\'u + f(u) em \'ômega\' x (0, T), u(x, 0) = \'fi\'(x) se x \'PERTENCE A\' \'ômega\', u(x, t) = 0 se x \'PERTENCE A\' \'delta\' \'ômega\', 0 < t < T, onde ­\'ômega\' é um domínio limitado em \'R POT.n\'com bordo \'C POT.2\', f é continuamente diferenciável com f(s) > 0, e \'fi\' é não-negativa e suave sobre \'ômega\'\'BARRA\' com \'fi\'=0 sobre \'delta\'\'ômega\'. Suponha que a única solução u(x,t) possui blow-up em tempo finito T < \'INFINITO\'. A questão que se coloca é: onde ocorre o blow-up? Neste trabalho provamos que: se \'ômega\'=\'B IND.R\'\'ESTÁ CONTIDO EM\'\'R POT. n\', então o blow-up ocorre apenas em r=0, Além disso, se f(u)=\'u POT.p\'p > 1, então u(r,t)\'< OU = \'C/\'r POT.2\'(\'gama\'-1) para qualquer 1 < \'gama\'< p, e assim \'limsup IND. t\'SETA\'T\'-||u(u.\'t)||q < \'INFINITO\'se q < n(p-1)/2. No caso não simétrico onde \'ômega\' é um domínio complexo, provamos que conjunto de blow-up é um subconjunto compacto de \'ômega\'. Se f(u)=\'u POT.p\', p > 1, então u(x,t)\'< OU = \'C/\'(T-t) POT. 1/p-1\' e, se n=1,2 ou se n\'< OU=\'3 p\'< OU=\'(n+2)/(n-2), então \'tau\'POT. \'beta\'u(x+\'Ksi\', T-\'tau\'\'SETA\'\'C IND. 0\' quando \'tau\'\'SETA\'\'0 POT. 1/2\'e \'C IND. 0\'= \'beta\'POT.\'beta\'\'onde \'beta\'= \'(p-1) POT. -1\'. As provas das estimativas essenciais para demonstração desses resultados são feitas utilizando o Princípio do Máximo / Consider the initial-boundary value problem \'u IND.t\'= \'delta\'u + f(u) in \'ômega\' x (0, T), u(x, 0) = \'fi\'(x) if x \'BELONGS\' \'ômega\', u(x, t) = 0 if x \'BELONGS \' \'\\PARTIAL\' \'ômega\', 0 < t < T, where ­\'ômega\' is a bounded domain in \'R POT.n\'with \'C POT.2\', f is continuously differentiable with f(s) > 0, and \'fi\' is nonnegative and smooth on \'ômega\'\'BARRA\' with \'fi\'=0 on \'\\PARTIIAL\'\'ômega\'. Assume that the unique solution u(x,t) blows up in finite time T < \'INFINITO\'. The question addressed is: where does the blow-up occur? In this work we prove: if \'ômega\'=\'B IND.R\'\'IS CONTAINED EM\'\'R POT. n\', then blow-up occurs only at r=0, Moreover, if f(u)=\'u POT.p\'p > 1, then u(r,t)\'< OU = \'C/\'r POT.2\'(\'gama\'-1) for any 1 < \'gama\'< p, and hence \'limsup IND. t\'SETA\'T\'-||u(u.\'t)||q < \'INFINITO\'se q < n(p-1)/2. In the nonsymmetric case where \'ômega\' is a convex domain, we prove that the blow-up set lies in a compact subset of \'ômega\'. If f(u)=\'u POT.p\', p > 1, then u(x,t)\'< OU = \'C/\'(T-t) POT. 1/p-1\' and, if n=1,2 or if n\'< OU=\'3 and p\'< OU=\'(n+2)/(n-2), then \'tau\'POT. \'beta\'u(x+\'Ksi\', T-\'tau\'\'SETA\'\'C IND. 0\' where \'tau\'\'SETA\'\'0 POT. 1/2\'e \'C IND. 0\'= \'beta\'POT.\'beta\'\'where \'beta\'= \'(p-1) POT. -1\'. Elementary applications of the Maximum Principle are used to prove the essential estimate for the proofs of these results.

Considere o problema de valor inicial e de fronteira \'u IND.t\'= \'delta\'u + f(u) em \'ômega\' x (0, T), u(x, 0) = \'fi\'(x) se x \'PERTENCE A\' \'ômega\', u(x, t) = 0 se x \'PERTENCE A\' \'delta\' \'ômega\', 0 < t < T, onde ­\'ômega\' é um domínio limitado em \'R POT.n\'com bordo \'C POT.2\', f é continuamente diferenciável com f(s) > 0, e \'fi\' é não-negativa e suave sobre \'ômega\'\'BARRA\' com \'fi\'=0 sobre \'delta\'\'ômega\'. Suponha que a única solução u(x,t) possui blow-up em tempo finito T < \'INFINITO\'. A questão que se coloca é: onde ocorre o blow-up? Neste trabalho provamos que: se \'ômega\'=\'B IND.R\'\'ESTÁ CONTIDO EM\'\'R POT. n\', então o blow-up ocorre apenas em r=0, Além disso, se f(u)=\'u POT.p\'p > 1, então u(r,t)\'< OU = \'C/\'r POT.2\'(\'gama\'-1) para qualquer 1 < \'gama\'< p, e assim \'limsup IND. t\'SETA\'T\'-||u(u.\'t)||q < \'INFINITO\'se q < n(p-1)/2. No caso não simétrico onde \'ômega\' é um domínio complexo, provamos que conjunto de blow-up é um subconjunto compacto de \'ômega\'. Se f(u)=\'u POT.p\', p > 1, então u(x,t)\'< OU = \'C/\'(T-t) POT. 1/p-1\' e, se n=1,2 ou se n\'< OU=\'3 p\'< OU=\'(n+2)/(n-2), então \'tau\'POT. \'beta\'u(x+\'Ksi\', T-\'tau\'\'SETA\'\'C IND. 0\' quando \'tau\'\'SETA\'\'0 POT. 1/2\'e \'C IND. 0\'= \'beta\'POT.\'beta\'\'onde \'beta\'= \'(p-1) POT. -1\'. As provas das estimativas essenciais para demonstração desses resultados são feitas utilizando o Princípio do Máximo / Consider the initial-boundary value problem \'u IND.t\'= \'delta\'u + f(u) in \'ômega\' x (0, T), u(x, 0) = \'fi\'(x) if x \'BELONGS\' \'ômega\', u(x, t) = 0 if x \'BELONGS \' \'\\PARTIAL\' \'ômega\', 0 < t < T, where ­\'ômega\' is a bounded domain in \'R POT.n\'with \'C POT.2\', f is continuously differentiable with f(s) > 0, and \'fi\' is nonnegative and smooth on \'ômega\'\'BARRA\' with \'fi\'=0 on \'\\PARTIIAL\'\'ômega\'. Assume that the unique solution u(x,t) blows up in finite time T < \'INFINITO\'. The question addressed is: where does the blow-up occur? In this work we prove: if \'ômega\'=\'B IND.R\'\'IS CONTAINED EM\'\'R POT. n\', then blow-up occurs only at r=0, Moreover, if f(u)=\'u POT.p\'p > 1, then u(r,t)\'< OU = \'C/\'r POT.2\'(\'gama\'-1) for any 1 < \'gama\'< p, and hence \'limsup IND. t\'SETA\'T\'-||u(u.\'t)||q < \'INFINITO\'se q < n(p-1)/2. In the nonsymmetric case where \'ômega\' is a convex domain, we prove that the blow-up set lies in a compact subset of \'ômega\'. If f(u)=\'u POT.p\', p > 1, then u(x,t)\'< OU = \'C/\'(T-t) POT. 1/p-1\' and, if n=1,2 or if n\'< OU=\'3 and p\'< OU=\'(n+2)/(n-2), then \'tau\'POT. \'beta\'u(x+\'Ksi\', T-\'tau\'\'SETA\'\'C IND. 0\' where \'tau\'\'SETA\'\'0 POT. 1/2\'e \'C IND. 0\'= \'beta\'POT.\'beta\'\'where \'beta\'= \'(p-1) POT. -1\'. Elementary applications of the Maximum Principle are used to prove the essential estimate for the proofs of these results.

On s’intéresse au phénomène d’explosion en temps fini dans les équations aux dérivées partielles paraboliques non linéaires, particulièrement au profil à l’explosion, des points de vue numérique et théorique. Dans la partie théorique, on s’intéresse au phénomène d’explosion en temps fini pour une classe d’équations semi linéaires de la chaleur perturbées fortement avec l’exposant sous-critique de Sobolev. Travaillant dans le cadre des variables auto-similaires, on obtient d’abord l’existence d’une fonctionnelle de Lyapunov, ce qui constitue une étape cruciale pour établir le taux d’explosion de la solution. Dans une seconde étape, on s’intéresse à la structure de la solution au voisinage du temps et du point d’explosion. On classifie tous les comportements asymptotiques possibles pour la solution quand elle s’approche de la singularité. Ensuite, on décrit les profils à l’explosion correspondant à ces comportements asymptotiques. Dans une troisième étape, on construit pour cette équation une solution qui explose en temps fini en un seul point avec un profil d’explosion prescrit. Cette construction s’appuie sur la réduction en dimension finie du problème et sur l’utilisation du théorème de l’indice pour conclure. Dans la partie numérique, on se propose de développer des méthodes afin de donner des réponses numériques à la question du profil à l’explosion pour certaines équations paraboliques, y compris le modèle de Ginzburg-Landau. Nous proposons deux méthodes. La première est l’algorithme de remise à l’échelle (rescaling) proposé par Bergeret Kohn en 1988, appliqué à des équations paraboliques satisfaisant une propriété d’invariance d’échelle. Cette propriété nous permet de faire un zoom de la solution quand elle est proche de la singularité, tout en gardant la même équation. Le principal avantage de cette méthode est sa capacité à donner une très bonne approximation numérique qui nous permet d’atteindre numériquement le profil à l’explosion. Le profil à l’explosion que l’on obtient numériquement est en bon accord avec le profil théorique. De plus, en considérant une équation de la chaleur non linéaire critique avec un terme de gradient non linéaire, avec peu de résultats théoriques, nous énonçons une conjecture sur le profil à l’explosion, grâce à nos simulations numériques. La deuxième méthode numérique s’appuie aussi sur un raffinement de maillage, dans l’esprit de l’algorithme de remise à l’échelle de Berger et Kohn. Cette méthode est applicable à une plus grande classe d’équations dont les solutions explosent en temps fini sans la propriété d’invariance d’échelle. / We are interested in finite-time blow-up phenomena arising in the study of Nonlinear Parabolic Partial Differential Equations, in particular in the blow-up profile, under the theoretical and numerical aspects. In the theoretical direction, we are interested in particular in finite-time blow-up phenomena for some class of strongly perturbed semilinear heat equations with Sobolev subcritical power nonlinearity. Working in the frameworkof similarity variables, we first derive a Lyapunov functional in similarity variables which is a crucial step to derive the blow-up rate of the solution. In a second step, we are interested in the structure of the solution near blow-uptime and point. We classify all possible asymptotic behaviors of the solution when it approaches to the singularity.Then we describe blow-up profiles corresponding to these asymptotic behaviors. In a third step, we construct for this equation a solution which blows up in finite time at only one blow-up point with a prescribed blow-up profile. The construction relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude. In the numerical direction, we intend to develop methods in order to give numerical answers to the question of the blow-up profile for some parabolic equations including the Ginzburg-Landau model. We propose two methods.The first one is the rescaling algorithm proposed by Berger and Kohn in 1988 applied to parabolic equations which are invariant under a scaling transformation. This scaling property allows us to make a zoom of the solution when it is close to the singularity, still keeping the same equation. The main advantage of this method is its ability to give a very good numerical approximation allowing to attain the numerical blow-up profile. The blow-up profile we obtain numerically is in good accordance with the theoretical one. Moreover, by applying the method to a critical nonlinear heat equation with a nonlinear gradient term, where almost nothing is known, we give a conjecture for its blow-up profile thanks to our numerical simulations. The second one is a new mesh-refinement method inspired by the rescaling algorithm of Berger and Kohn, which is applicable to more general equations, in particular those with no scaling invariance.

En el presente trabajo, estudiamos sobre la reducción de singularidades de un campo vectorial analítico en C2. Presentaremos un resultado de J. Mattei and R. Moussu, esté resultado prueba que después de un número finito de blow-ups las singularidades son simples. PALABRAS CLAVES: Campos vectoriales analíticos, Singularidades, Blow-up. / --- In this work, we study about reduction of singularities of analytic vector fields on C2 . We will present a result of J. Mattei and R. Moussu [7], that result showed that after a finite number of blow-ups the singularities are simple. KEYWORDS: Analytic vector fields, Singularities, Blow-up. / Tesis

In 2002, Feldman, Ilmanen, and Knopf constructed the first example of a non-trivial (i.e. non-constant curvature) complete non-compact shrinking soliton, and conjectured that it models a Ricci flow singularity forming on a closed four-manifold. In this thesis, we confirm their conjecture and, as a consequence, show that limits of blow-ups of Ricci flow singularities on closed four-dimensional manifolds do not necessarily have non-negative Ricci curvature. / text

A long-standing problem in cobordism theory has been to find convenient manifolds to represent cobordism classes. For example, in the late 1950's, Hirzebruch asked which complex cobordism classes can be represented by smooth connected algebraic varieties. This question is still open. Progress can be made on this and related problems by studying certain convenient connected algebraic varieties, namely smooth projective toric varieties. The primary focus of this dissertation is to determine which complex cobordism classes can be represented by smooth projective toric varieties. A complete answer is given up to dimension six, and a partial answer is described in dimension eight. In addition, the role of smooth projective toric varieties in the polynomial ring structure of complex cobordism is examined. More specifically, smooth projective toric varieties are constructed as polynomial ring generators in most dimensions, and evidence is presented suggesting that a smooth projective toric variety can be chosen as a polynomial generator in every dimension. Finally, toric varieties with an additional fiber bundle structure are used to study some manifolds in oriented cobordism. In particular, manifolds with certain fiber bundle structures are shown to all be cobordant to zero in the oriented cobordism ring.

L'éclatement est une transformation jouant un rôle important en géométrie, car il permet de résoudre des singularités, de relier des variétés birationnellement équivalentes, et de construire des variétés possédant des propriétés inédites. Ce mémoire présente d'abord l'éclatement tel que développé en géométrie algébrique classique. Nous l'étudierons pour le cas des variétés affines et (quasi-)projectives, en un point, et le long d'un idéal et d'une sous-variété. Nous poursuivrons en étudiant l'extension de cette construction à la catégorie différentiable, sur les corps réels et complexes, en un point et le long d'une sous-variété. Nous conclurons cette section en explorant un exemple de résolution de singularité. Ensuite nous passerons à la catégorie symplectique, où nous ferons la même chose que pour le cas différentiable complexe, en portant une attention particulière à la forme symplectique définie sur la variété. Nous terminerons en étudiant un théorème dû à François Lalonde, où l'éclatement joue un rôle clé dans la démonstration. Ce théorème affirme que toute 4-variété fibrée par des 2-sphères sur une surface de Riemann, et différente du produit cartésien de deux 2-sphères, peut être équipée d'une 2-forme qui lui confère une structure symplectique réglée par des courbes holomorphes par rapport à sa structure presque complexe, et telle que l'aire symplectique de la base est inférieure à la capacité de la variété. La preuve repose sur l'utilisation de l'éclatement symplectique. En effet, en éclatant symplectiquement une boule contenue dans la 4-variété, il est possible d'obtenir une fibration contenant deux sphères d'auto-intersection -1 distinctes: la pré-image du point où est fait l'éclatement complexe usuel, et la transformation propre de la fibre. Ces dernières sont dites exceptionnelles, et donc il est possible de procéder à l'inverse de l'éclatement - la contraction - sur chacune d'elles. En l'accomplissant sur la deuxième, nous obtenons une variété minimale, et en combinant les informations sur les aires symplectiques de ses classes d'homologies et de celles de la variété originale nous obtenons le résultat. / The blow-up is a transformation which plays an important role in geometry, because it can be used to resolve singularities, relate birationally equivalent varieties, and construct varieties with new properties. This thesis first presents blowing-up as developped in classical algebraic geometry. We will study it in the case of affine and (quasi-)projective varieties, on a point and along an ideal and a subvariety. Then a discussion about its extension to the differential category will be carried out, over the real and complex fields, on a point and along a submanifold. An example of a resolution of singularity will then follow. Subsequently we will discuss blowing-up in the symplectic category, where we will do the same as for complex manifolds, paying careful attention to the symplectic form. To conclude, we will study a theorem by François Lalonde, where the symplectic blow-up plays a major part in proof. This theorem states that any 4-variety fibered by 2-spheres over a Riemann surface, and different than the Cartesian product of two 2-spheres, can be equiped with a 2-form giving it a symplectic structure ruled by curves that are holomorphic with respect to its almost-complex structure, and such that the symplectic area of the base is smaller that the capacity of the variety. In the proof, we blow up a ball in the 4-variety, and obtain a fibration containing two distinct spheres with a self-intersection equal to -1: the pre-image of the point where the usual complex blow-up is done, and the proper transform of the fiber. These two are exceptional, so it is possible to do the inverse operation - the blow down - on each of them. By blowing down the latter, we get a minimal variety, and by combining information about the symplectic area of its homology classes and of those of the original variety, we obtain the result.

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