本研究中討論了非線性微分方程式之解的正則性。在這之中發現了一些有趣的現象，得到了方程式解可以做任意次的微分，並且得到對該解任意次微分後其值趨近到無限大時之爆破速率、爆破常數及當其值遞減至零時的爆破速率、爆破常數。 / In this paper we work with the regularity of solutions for the non-linear ordinary differential equation u''-u^p=0 for some well-defined functions u^p. We have found some interesting phenomena, u belongs to C^q for any q in positive integer, blow-up constant, blow-up rate, null point and decay rate of u^(n) are obtained in this work, through that we get the characterization for these equations in this case.

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).

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-17T15:40:30Z No. of bitstreams: 1 arquivototal.pdf: 1244185 bytes, checksum: d60955ede563305b1f641dd53d7154a7 (MD5) / Made available in DSpace on 2017-08-17T15:40:30Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1244185 bytes, checksum: d60955ede563305b1f641dd53d7154a7 (MD5) Previous issue date: 2017-03-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In thisworkwestudytheasymptoticbehaviorofthesolutionsofpartialdi erential equations slowlynon-dissipative,wewillgiveabriefhistoricaloverviewonthetheme, wewillpresentanintroductiontothetheoryofsemigroupsofboundedlinearoperators in Banachspaces,fractionalpowersofsectoroperatorsandresultsontheexistenceand uniqueness ofsolutionsofabstractCauchysemilinearproblemsoftheparabolictype. / Neste trabalhoestudamosocomportamentoassint oticodassolu c~oesdeequa c~oes diferenciais parciaislentamenten~aodissipativas,daremosumbreveapanhadohist orico sobre otema,apresentaremosumaintrodu c~ao ateoriadesemigruposdeoperadores lineares limitadosemespa cosdeBanach,pot^enciasfracion ariasdeoperadoressetoriais e resultadossobreexist^enciaeunicidadedesolu c~oesdeproblemasabstratosdeCauchy semilineares dotipoparab olico.

Relaxation oscillations are highly non-linear oscillations, which appear to feature many important biological phenomena such as heartbeat, neuronal activity, and population cycles of predator-prey type. They are characterized by repeated switching of slow and fast motions and occur naturally in singularly perturbed ordinary differential equations, which exhibit dynamics on different time scales. Traditionally, slow-fast systems and the related oscillatory phenomena -- such as relaxation oscillations -- have been studied by the method of the matched asymptotic expansions, techniques from non-standard analysis, and recently a more qualitative approach known as geometric singular perturbation theory. It turns out that relaxation oscillations can be found in a more general setting; in particular, in slow-fast systems, which are not written in the standard form. Systems in which separation into slow and fast variables is not given a priori, arise frequently in applications. Many of these systems include additionally various parameters of different orders of magnitude and complicated (non-polynomial) non-linearities. This poses several mathematical challenges, since the application of singular perturbation arguments is not at all straightforward. For that reason most of such systems have been studied only numerically guided by phase-space analysis arguments or analyzed in a rather non-rigorous way. It turns out that the main idea of singular perturbation approach can also be applied in such non-standard cases. This thesis is concerned with the application of concepts from geometric singular perturbation theory and geometric desingularization based on the blow-up method to the study of relaxation oscillations in slow-fast systems beyond the standard form. A detailed geometric analysis of oscillatory mechanisms in three mathematical models describing biochemical processes is presented. In all the three cases the aim is to detect the presence of an isolated periodic movement represented by a limit cycle. By using geometric arguments from the perspective of dynamical systems theory and geometric desingularization based on the blow-up method analytic proofs of the existence of limit cycles in the models are provided. This work shows -- in the context of non-trivial applications -- that the geometric approach, in particular the blow-up method, is valuable for the understanding of the dynamics of systems with no explicit splitting into slow and fast variables, and for systems depending singularly on several parameters.

The generalized inviscid Proudman-Johnson equation serves as a model for n-dimensional incompressible Euler flow, gas dynamics, high-frequency waves in shallow waters, and orientation of waves in a massive director field of a nematic liquid crystal. Furthermore, the equation also serves as a tool for studying the role that the natural fluid processes of convection and stretching play in the formation of spontaneous singularities, or of their absence. In this work, we study blow-up, and blow-up properties, in solutions to the generalized, inviscid Proudman-Johnson equation endowed with periodic or Dirichlet boundary conditions. More particularly,regularity of solutions in an Lp setting will be measured via a direct approach which involves the derivation of representation formulae for solutions to the problem. For a real parameter lambda, several classes of initial data are considered. These include the class of smooth functions with either zero or nonzero mean, a family of piecewise constant functions, and a large class of initial data with a bounded derivative that is, at least, continuous almost everywhere and satisfies Holder-type estimates near particular locations in the domain. Amongst other results, our analysis will indicate that for appropriate values of the parameter, the curvature of the data in a neighborhood of these locations is responsible for an eventual breakdown of solutions, or their persistence for all time. Additionally, we will establish a nontrivial connection between the qualitative properties of L-infinity blow-up in ux, and its Lp regularity. Finally, for smooth and non-smooth initial data, a special emphasis is made on the study of regularity of stagnation point-form solutions to the two and three dimensional incompressible Euler equations subject to periodic or Dirichlet boundary conditions.

We consider the Cauchy problem for the focusing energy critical NLS with inverse square potential. The energy of the solution, which consists of the kinetic energy and potential energy, is conserved for all time. Due to the focusing nature, solution with arbitrary energy may exhibit various behaviors: it could exist globally and scatter like a free evolution, persist like a solitary wave, blow up at finite time, or even have mixed behaviors. Our goal in this thesis is to fully characterize the solution when the energy is below or at the level of the energy of the ground state solution $W_a$. Our main result contains two parts. First, we prove that when the energy and kinetic energy of the initial data are less than those of the ground state solution, the solution exists globally and scatters. Second, we show a rigidity result at the level of ground state solution. We prove that among all solutions with the same energy as the ground state solution, there are only two (up to symmetries) solutions $W_a^+, W_a^-$ that are exponential close to $W_a$ and serve as the threshold of scattering and blow-up. All solutions with the same energy will blow up both forward and backward in time if they go beyond the upper threshold $W_a^+$; all solutions with the same energy will scatter both forward and backward in time if they fall below the lower threshold $W_a^-$. In the case of NLS with no potential, this type of results was first obtained by Kenig-Merle \cite{R: Kenig focusing} and Duyckaerts-Merle \cite{R: D Merle}. However, as the potential has the same scaling as $\Delta$, one can not expect to extend their results in a simple perturbative way. We develop crucial spectral estimates for the operator $-\Delta+a/|x|^2$, we also rely heavily on the recent understanding of the operator $-\Delta+a/|x|^2$ in \cite{R: Harmonic inverse KMVZZ}.

This thesis contributes towards the maximal-in-time well-posedness theory of three nonlinear dispersive partial differential equations (PDEs). We are interested in questions that extend beyond the usual well-posedness theory: what is the ultimate fate of solutions? How does Hamiltonian structure influence PDE dynamics? How does randomness, within the PDE or the initial data, interact with well-posedness of the Cauchy problem? The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of surface water waves. We prove a mass concentration property for H^1-solutions, analogous to the one known for the L^2-critical nonlinear Schrodinger equation. We also prove a mass concentration result for L^2-solutions. The second topic of this thesis is the invariance of the Gibbs measure for the (gauge transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure supported on H^s for s<1/2, and local solutions to the quartic KdV cannot be obtained below H^{1/2} by using the standard fixed point method. We exhibit nonlinear smoothing when the initial data are randomized, and establish almost sure local well-posedness for the (gauge transformed) quartic KdV below H^{1/2}. Then, using the invariance of the Gibbs measure for the finite-dimensional system of ODEs given by projection onto the first N>0 modes of the trigonometric basis, we extend the local solutions of the (gauge transformed) quartic KdV to global solutions, and prove the invariance of the Gibbs measure under the flow. Inverting the gauge, we establish almost sure global well-posedness of the (ungauged) periodic quartic KdV below H^{1/2}. The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equation. This equation is studied as a toy model for the stochastic Burgers equation, which appears in the description of a randomly growing interface. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This thesis provides a result in this direction: after smoothing the additive noise (by a fractional derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.

A first goal of this thesis is to contribute to the knowledge of cohomological properties of non-standard multigraded modules. In particular we study the Hilbert function of a non-standard multigraded module, the asymptotic depth of the homogeneous components of a multigraded module and the asymptotic depth of the Veronese modules. To reach our purposes, we generalize some cohomological invariants to the non-standard multigraded case and we study properties on the vanishing of local cohomology modules. In particular we study the generalized depth of a multigraded module.In chapters 2, 3 and 4, we consider multigraded rings S, finitely generated over the local ring S0 by elements of degrees g1,.,gr with gi=(g1i,.,gii,.,0) non-negative integral vectors and gii not zero for i=1,.,r. In Chapter 2, we prove that the Hilbert function of a multigraded S-module is quasi-polynomial in a cone of N^r. Moreover the Grothendieck-Serre formula is satisfied in our situation as well.In Chapter 3, using the quasi-polynomial behavior of the Hilbert function of the Koszul homology modules of a multigraded S-module M with respect to a system of generators of the maximal ideal of S0, we can prove that the depth of the homogeneous components of M is constant for degrees in a subnet of a cone of N^r defined by g1,.,gr. In some cases we can assure constant depth in all the cone. By considering the multigraded blow-up algebras associated to ideals I1,.,Ir in a Noetherian local ring (R,m), we can prove that the depth of R/I1^n1.Ir^nr is constant for n1,.,nr large enough.In Chapter 4, we study the depth of (a,b)-Veronese modules for a, b large enough. In particular we prove that in almost-standard case (i.e. the degrees of the generators are positive multiples of the canonical basis) with S0 a quotient of a regular local ring, this depth is constant for a, b in some regions of N^r. To reach this result we need a previous study about Veronese modules and about the vanishing of local cohomology modules. In particular we prove that, in the moregeneral case, if S0 is a quotient of a regular local ring, the generalized depth is invariant by taking Veronese transforms. Moreover in the almost-standard case the generalized depth coincides with the index of finite graduation of the local cohomology modules with respect to the homogeneous maximal ideal.A second goal of the thesis is the study of the depth of blow-up algebras associated to an ideal. In Chapter 5 we obtain refined versions of some conjectures on the depth of the associated graded ring of an ideal. By using certain non-standard bigraded structures, the integers that appear in Guerrieri's Conjecture and in Wang's Conjecture can be interpreted as a multiplicities of some bigraded modules. In particular we have given an answer to the question formulated by A. Guerrieri and C. Huneke in 1993. We have proved that given an m-primary ideal I in a Cohen-Macaulay local ring (R,m) of dimension d>0 with minimal reduction J, assuming that the lengths of the homogeneous components of the Valabrega-Valla module of I and J are less than or equal to 1, then the depth of the associated graded ring of I is greater than or equal to d-2.Finally, in Chapter 6, the study of the Hilbert function of certain submodules of the bigraded modules studied before, allows us to prove some cases in which the Hilbert function of an m-primary ideal in a one-dimensional Cohen-Macaulay local ring is non-decreasing. / CATALÀ: TÍTOL DE LA TESI: "Estructures Multigraduades i la Profunditat d'Àlgebres de Blow-up"TEXT DEL RESUM:Un primer objectiu d'aquesta tesi és contribuir al coneixement de propietats cohomològiques de mòduls multigraduats no-estàndard. En particular estudiem la funció de Hilbert d'un mòdul multigraduat no-estàndard, la profunditat asimptòtica de les components homogènies d'un mòdul multigraduat i la profunditat asimptòtica dels mòduls de Veronese. Per a això, generalitzem alguns invariants cohomològics en el cas multigraduat no-estàndard i estudiem propietats d'anul·lació de mòduls de cohomologia local. En particular estudiem la profunditat generalitzada d'un mòdul multigraduat.En els capítols 2, 3 i 4, considerem anells multigraduats S finitament generats sobre l'anell local S0 per elements de graus g1,...,gr amb gi=(g1i,...,gii,...,0) vectors enters no-negatius i gii no nul per a i=1,...,r. Al Capítol 2, demostrem que la funció de Hilbert d'un S-mòdul multigraduat és quasi-polinòmica en un con de N^r. A més es satisfà la fórmula de Grothendieck-Serre en la nostra situació.Al Capítol 3, utilitzant el comportament quasi-polinòmic de la funció de Hilbert dels mòduls d'homologia de Koszul d'un S-mòdul M multigraduat respecte d'un sistema de generadors de l'ideal maximal de S0, podem demostrar que la profunditat de les components homogènies de M és constant per a graus en una subxarxa d'un con de N^r definit per g1,...,gr. En alguns casos es pot assegurar profunditat constant en tot un con. Considerant els anells de blow-up multigraduats associats a ideals I1,...,Ir en un anell local Noetherià (R,m), podem demostrar que la profunditat de R/I1^n1...Ir^nr és constant per a n1,...,nr prou grans.Al Capítol 4, estudiem la profunditat dels mòduls de (a,b)-Veronese per a a,b prou grans. En particular demostrem que en el cas quasi-estàndard (i.e. amb generadors de graus múltiples positius de la base canònica) amb S0 quocient d'un anell local regular, aquesta profunditat és constant per a a,b en certes regions de N^r. Per arribar a aquest resultat ens cal un estudi previ dels mòduls de Veronese i de l'anul·lació de mòduls de cohomologia local. En particular demostrem que, en el cas més general, si S0 és quocient d'un anell local regular, la profunditat generalitzada és invariant per transformacions Veronese. A més en el cas quasi-estàndard la profunditat generalitzada coincideix amb l'índex de graduació finita dels mòduls de cohomologia local respecte de l'ideal homogeni maximal.Un segon objectiu de la tesi és l'estudi de la profunditat de les àlgebres de blow-up associades a un ideal. Al Capítol 5 s'obtenen versions refinades de conjectures sobre la profunditat de l'anell graduat associat a un ideal. Utilitzant algunes estructures bigraduades no-estàndard, es poden interpretar els enters que apareixen a la Conjectura de Guerrieri i a la Conjectura de Wang com a multiplicitats de mòduls bigraduats. En particular hem pogut donar resposta a una pregunta formulada per A. Guerrieri i C. Huneke al 1993. Hem demostrat que donat un ideal I m-primari en un anell local (R,m) Cohen-Macaulay de dimensió d>0 amb reducció minimal J, suposant que les longituds de les components homogènies del mòdul de Valabrega-Valla de I i J siguin menors o iguals que 1, aleshores la profunditat de l'anell graduat associat a I és major o igual que d-2.Finalment, al Capítol 6, l'estudi de la funció de Hilbert de certs submòduls dels mòduls bigraduats estudiats anteriorment, permet provar alguns casos en què la funció de Hilbert d'un ideal m-primari en un anell local Cohen-Macaulay de dimensió 1, és no decreixent.

Plusieurs problèmes d'optimisation liés à la théorie du transport optimal, concernant aussi des critères de concentration, sont étudiés. Il s'agit, pour ce qui est des primiers chapitres, de la minimization de fonctionnelles definies sur les mesures marginales du porblème de transport, en demandant que l'une soit concentrée et l'autre diffusée, alors que les deux doivent être proche au sense du transport de masse. D'autres chapitres portent sur des modèles différentes qui considèrent la concentration des parcours suivis par les particules lors du mouvement, en donnant des effets de congestion ou branchement. Plusieurs problèmes font apparaître des structures de dimension 1 (reseaux, supports rectifiables de mesures vectorielles, ensemble sous contraintes de longueur...) et leur régularité (blow-up) est étudiée dans les deux derniers chapitres. Les modèles viennent dans la majorité des cas de possibles applications à la planification urbaine, la biologie (arbres, feuilles et système sanguin), la géophysique (bassins fluviaux) et la mécanique des fluides. La thèse a été écrite sous la direction du Prof. Buttazzo et soutenue à l'Ecole Normale Supérieure de Pise.

Ramsey Theorem, in the most simple form, states that if we are given a positive integer l, there exists a minimal integer r(l), called the Ramsey number, such any partition of the edges of K_r(l) into two sets, i.e. a 2-coloring, yields a copy of K_l contained entirely in one of the partitioned sets, i.e. a monochromatic copy of Kl. We prove an extension of Ramsey's Theorem, in the more general form, by replacing complete graphs by multipartite graphs in both senses, as the partitioned set and as the desired monochromatic graph. More formally, given integers l and k, there exists an integer p(m) such that any 2-coloring of the edges of the complete multipartite graph K_p(m);r(k) yields a monochromatic copy of K_m;k . The tools that are used to prove this result are the Szemeredi Regularity Lemma and the Blow Up Lemma. A full proof of the Regularity Lemma is given. The Blow-Up Lemma is merely stated, but other graph embedding results are given. It is also shown that certain embedding conditions on classes of graphs, namely (f , ?) -embeddability, provides a method to bound the order of the multipartite Ramsey numbers on the graphs. This provides a method to prove that a large class of graphs, including trees, graphs of bounded degree, and planar graphs, has a linear bound, in terms of the number of vertices, on the multipartite Ramsey number.