We study blowup of solutions of one-dimensional nonlinear heat equations (NLH). We consider two cases: a power nonlinearity and initial conditions having two equal absolute maxima and a polynomial nonlinearity and initial conditions having a single global maximum. We show in both cases that for a certain open set of initial conditions solutions of the NLH blow up in finite time and we find asymptotical behavior of blowup frofiles. In the first case the blowup occurs at two points while in the second case, at one point.

We study blowup of solutions of one-dimensional nonlinear heat equations (NLH). We consider two cases: a power nonlinearity and initial conditions having two equal absolute maxima and a polynomial nonlinearity and initial conditions having a single global maximum. We show in both cases that for a certain open set of initial conditions solutions of the NLH blow up in finite time and we find asymptotical behavior of blowup frofiles. In the first case the blowup occurs at two points while in the second case, at one point.

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-06-13T17:22:58Z No. of bitstreams: 1 WashingtonMenezes.pdf: 349401 bytes, checksum: 40e24e24fcc72911a8a30e2a9d7b0f1b (MD5) / Made available in DSpace on 2017-06-13T17:22:58Z (GMT). No. of bitstreams: 1 WashingtonMenezes.pdf: 349401 bytes, checksum: 40e24e24fcc72911a8a30e2a9d7b0f1b (MD5) Previous issue date: 2016-08-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / In this work, we introduce results of local and global solution for a heat equation in RN with nonlocal nonlinearity in time. / Neste trabalho, apresentamos resultados de existência local e global para uma equação do calor em RN com não-linearidade não local no tempo.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation, we will use the techniques established in [2] to analyze the existence and uniqueness of classical solution to the nonlinear heat equation in ; provided u0 2 Lq() with some conditions on 1 q 1, where RN is a smooth bounded domain and p > 1. / Nesta dissertação, vamos utilizar as técnicas vistas em [2] para analisar a existência e unicidade de solução clássica para a equação do calor não-linear quando uo E Lq com algumas condições sobre 1<q<, onde RN é um domínio limitado suave e p>1.

Hauptthema der Dissertation ist die Analysis eines nichtlinearen, gekoppelten Systems partieller Differentialgleichungen (PDG), das in der Modellierung der Kristallzüchtung aus der Schmelze mit Magnetfeldern vorkommt. Die zu beschreibenden Phenomäne sind einerseits der im elektromagnetisch geheizten Schmelzofen erfolgende Wärmetransport (Wärmeleitung, -konvektion und -strahlung), und andererseits die Bewegung der Halbleiterschmelze unter dem Einfluss der thermischen Konvektion und der angewendeten elektromagnetischen Kräfte. Das Modell besteht aus den Navier-Stokeschen Gleichungen für eine inkompressible Newtonsche Flüssigkeit, aus der Wärmeleitungsgleichung und aus der elektrotechnischen Näherung des Maxwellschen Systems. Wir erörtern die schwache Formulierung dieses PDG Systems, und wir stellen ein Anfang-Randwertproblem auf, das die Komplexität der Anwendung widerspiegelt. Die Hauptfrage unserer Untersuchung ist die Wohlgestelltheit dieses Problems, sowohl im stationären als auch im zeitabhängigen Fall. Wir zeigen die Existenz schwacher Lösungen in geometrischen Situationen, in welchen unstetige Materialeigenschaften und nichtglatte Trennfläche auftreten dürfen, und für allgemeine Daten. In der Lösung zum zeitabhängigen Problem tritt ein Defektmaß auf, das ausser der Flüssigkeit im Rand der elektrisch leitenden Materialien konzentriert bleibt. Da eine globale Abschätzung der im Strahlungshohlraum ausgestrahlten Wärme auch fehlt, rührt ein Teil dieses Defektmaßes von der nichtlokalen Strahlung her. Die Eindeutigkeit der schwachen Lösung erhalten wir nur unter verstärkten Annahmen: die Kleinheit der gegebenen elektrischen Leistung im stationären Fall, und die Regularität der Lösung im zeitabhängigen Fall. Regularitätseigenschaften wie die Beschränktheit der Temperatur werden, wenn auch nur in vereinfachten Situationen, hergeleitet: glatte Materialtrennfläche und Temperaturunabhängige Koeffiziente im Fall einer stationären Analysis, und entkoppeltes, zeitharmonisches Maxwell für das transiente Problem. / The present PhD thesis is devoted to the analysis of a coupled system of nonlinear partial differential equations (PDE), that arises in the modeling of crystal growth from the melt in magnetic fields. The phenomena described by the model are mainly the heat-transfer processes (by conduction, convection and radiation) taking place in a high-temperatures furnace heated electromagnetically, and the motion of a semiconducting melted material subject to buoyancy and applied electromagnetic forces. The model consists of the Navier-Stokes equations for a newtonian incompressible liquid, coupled to the heat equation and the low-frequency approximation of Maxwell''s equations. We propose a mathematical setting for this PDE system, we derive its weak formulation, and we formulate an (initial) boundary value problem that in the mean reflects the complexity of the real-life application. The well-posedness of this (initial) boundary value problem is the mainmatter of the investigation. We prove the existence of weak solutions allowing for general geometrical situations (discontinuous coefficients, nonsmooth material interfaces) and data, the most important requirement being only that the injected electrical power remains finite. For the time-dependent problem, a defect measure appears in the solution, which apart from the fluid remains concentrated in the boundary of the electrical conductors. In the absence of a global estimate on the radiation emitted in the cavity, a part of the defect measure is due to the nonlocal radiation effects. The uniqueness of the weak solution is obtained only under reinforced assumptions: smallness of the input power in the stationary case, and regularity of the solution in the time-dependent case. Regularity properties, such as the boundedness of temperature are also derived, but only in simplified settings: smooth interfaces and temperature-independent coefficients in the case of a stationary analysis, and, additionally for the transient problem, decoupled time-harmonic Maxwell.

Dans cette thèse, nous étudions l’équation parabolique non linéaire ∂ t u = ∆u + a |x|⎺⥾ |u|ᵅ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, ⍺ ∈ R, α > 0, 0 < Ƴ < min(2,N) et avec une donnée initiale u(0) = φ. On établit l’existence et l’unicité locale dans Lq(Rᴺ) et dans Cₒ(Rᴺ). En particulier, la valeur q = N ⍺/(2 − γ) joue un rôle critique. Pour ⍺ > (2 − γ)/N, on montre l’existence de solutions auto-similaires globales avec données initiales φ(x) = ω(x) |x|−(2−γ)/⍺, où ω ∈ L∞(Rᴺ) homogène de degré 0 et ||ω||∞ est suffisamment petite. Nous montrons ainsi que si φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺ pour |x| grande, alors la solution est globale et asymptotique dans L∞(Rᴺ) à une solution auto-similaire de l’équation non linéaire. Tandis que si φ(x)∼ω(x) |x| (x)|x|−σ pour des |x| grandes avec (2 − γ)/⍺ < σ < N, alors la solution est globale, mais elle est asymptotique dans L∞(Rᴺ) à eᵗ∆(ω(x) |x|−σ). L’équation avec un potentiel plus général, ∂ t u = ∆u + V(x) |u|ᵅ u, V(x) |x |⥾ ∈ L∞(Rᴺ), est également étudiée. En particulier, pour des données initiales φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺, |x| grande, nous montrons que le comportement à grand temps est linéaire si V est à support compact au voisinage de l’origine, alors qu’il est non linéaire si V est à support compact au voisinage de l’infini. Nous étudions également le système non linéaire ∂ t u = ∆u + a |x|⎺⥾ |v|ᴾ⎺¹v, ∂ t v = ∆v + b |x|⎺ ᴾ |u|q⎺¹ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, a,b ∈ R, 0 < y < min(2,N)? 0 < p < min(2,N), p,q > 1. Sous des conditions sur les paramètres p, q, γ et ρ nous montrons l’existence et l’unicité de solutions globales avec données initiales petites par rapport à certaines normes. En particulier, on montre l’existence de solutions auto-similaires avec donnée initiale Φ = (φ₁, φ₂), où φ₁, φ₂ sont des données initiales homogènes. Nous montrons également que certaines solutions globales sont asymptotiquement auto-similaires. Comme deuxième objectif, nous considérons l’équation de la chaleur non linéaire ut = ∆u + |u|ᴾ⎺¹u - |u| q⎺¹u, avec t ≥ 0 et x ∈ Ω, la boule unité de Rᴺ, N ≥ 3, avec des conditions aux limites de Dirichlet. Soit h une solution stationnaire à symétrie radiale avec changement de signe de (E). On montre que la solution de (E) avec donnée initiale λh explose en temps fini si |λ − 1| > 0 est suffisamment petit et si 1 < q < p < Ps = N+2/N−2 et p suffisamment proche de Ps. Ceci prouve que l’ensemble des données initiales pour lesquelles la solution est globale n’est pas étoilé au voisinage de 0. / In this thesis, we study the nonlinear parabolic equation ∂ t u = ∆u + a |x|⎺⥾ |u|ᵅ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, ⍺ ∈ R, α > 0, 0 < Ƴ < min(2,N) and with initial value u(0) = φ. We establish local well-posedness in Lq(Rᴺ) and in Cₒ(Rᴺ). In particular, the value q = N ⍺/(2 − γ) plays a critical role.For ⍺ > (2 − γ)/N, we show the existence of global self-similar solutions with initial values φ(x) = ω(x) |x|−(2−γ)/⍺, where ω ∈ L∞(Rᴺ) is homogeneous of degree 0 and ||ω||∞ is sufficiently small. We then prove that if φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺ for |x| large, then the solution is global and is asymptotic in the L∞-norm to a self-similar solution of the nonlinear equation. While if φ(x)∼ω(x) |x| (x)|x|−σ for |x| large with (2 − γ)/α < σ < N, then the solution is global but is asymptotic in the L∞-norm toe t(ω(x) |x|−σ). The equation with more general potential, ∂ t u = ∆u + V(x) |u|ᵅ u, V(x) |x |⥾ ∈ L∞(Rᴺ), is also studied. In particular, for initial data φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺, |x| large , we show that the large time behavior is linear if V is compactly supported near the origin, while it is nonlinear if V is compactly supported near infinity. we study also the nonlinear parabolic system ∂ t u = ∆u + a |x|⎺⥾ |v|ᴾ⎺¹v, ∂ t v = ∆v + b |x|⎺ ᴾ |u|q⎺¹ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, a,b ∈ R, 0 < y < min(2,N)? 0 < p < min(2,N), p,q > 1. Under conditions on the parameters p, q, γ and ρ we show the existence and uniqueness of global solutions for initial values small with respect of some norms. In particular, we show the existence of self-similar solutions with initial value Φ = (φ₁, φ₂), where φ₁, φ₂ are homogeneous initial data. We also prove that some global solutions are asymptotic for large time to self-similar solutions. As a second objective we consider the nonlinear heat equation ut = ∆u + |u|ᴾ⎺¹u - |u| q⎺¹u, where t ≥ 0 and x ∈ Ω, the unit ball of Rᴺ, N ≥ 3, with Dirichlet boundary conditions. Let h be a radially symmetric, sign-changing stationary solution of (E). We prove that the solution of (E) with initial value λ h blows up in finite time if |λ − 1| > 0 is sufficiently small and if 1 < q < p < Ps = N+2/N−2 and p sufficiently close to Ps. This proves that the set of initial data for which the solution is global is not star-shaped around 0.