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.

In this paper we construct explicit examples of solutions to certain nonlinear wave equations. These semilinear equations are the simplest equations known to possess localized solitary waves in more that one spatial dimension. We construct explicit localized standing wave solutions, which generate multidimensional localized traveling solitary waves under the action of velocity boosts. We study the case of two spatial dimensions and a piecewise-linear nonlinearity. We obtain a large subset of the infinite family of standing waves, and we exhibit several interesting features of the family. Our solutions include solitary waves that carry nonzero angular momenta in their rest frames. The spatial profiles of these solutions also furnish examples of symmetry breaking for nonlinear elliptic equations.

Made available in DSpace on 2015-05-15T11:46:05Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 813665 bytes, checksum: 8aa09df2661d8ea4c0561ebad8cd9584 (MD5) Previous issue date: 2012-07-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The goal of our work is to prove the existence of solutions to a class of semilinear elliptic equations in a bounded domain, involving concave-convex type nonlinearities. We use a variety of methods to and these solutions, such as Mountain Pass Theorem, Ekeland's Variational Principle, Lagrange Multipliers Theorem, Nehari Manifold and sub and supersolution method. / O objetivo da nossa dissertação é provar a existência de soluções para uma classe de equações elípticas semilineares em um domínio limitado, envolvendo não linearidades do tipo côncavo-convexas. Mostraremos alguns casos diferentes e métodos diversificados para encontrar tais soluções, usando o Teorema do Passo da Montanha, o Princípio Variacional de Ekeland, Teorema dos Multiplicadores de Lagrange, a Variedade de Nehari e sub e supersolução.

Cette thèse est consacrée à l’étude des équations aux dérivées partielles stochastiques de type parabolique. Dans la première partie nous démontrons de nouveaux résultats concernant l’existence et l’unicité de solutions variationnelles globales et locales à des problèmes avec des conditions aux bords de type Neumann pour une classe d’équations aux dérivées partielles stochastiques non-autonomes. Les équations que nous considérons sont définies sur des domaines non bornés de l’espace euclidien qui satisfont à certaines conditions géométriques, et sont dirigées par un bruit multiplicatif dérivé d’un processus de Wiener fractionnaire infini-dimensionnel caractérisé par une suite de paramètres de Hurst H = (Hi) i ∈ N+ ⊂ (1/2,1). Ces paramètres sont en fait soumis à d’autres contraintes intimement liées à la nature de la non-linéarité dans le terme stochastique des équations, et au choix des espaces fonctionnels dans lesquels le problème à résoudre est bien posé. Notre méthode de preuve repose essentiellement sur des arguments d’injections compactes. Dans la seconde partie, nous étudions la possibilité de l’explosion de solutions d’une classe d’équations aux dérivées partielles stochastiques semi-linéaire avec des conditions aux bords de type Dirichlet, perturbées par un mélange d’un mouvement brownien et d’un mouvement brownien fractionnaire et dirigées par une classe d’opérateurs différentiels non autonomes contenant des processus de diffusions et des processus de Lévy. Notre but est de comprendre l’influence de la partie stochastique et de l’opérateur différentiel sur le comportement d’explosion des solutions. En particulier, nous donnons des expressions explicites pour des bornes inférieures et supérieures du temps de l’explosion de la solution, et des conditions suffisantes pour l’existence d’une solution globale positive. Nous estimons également la probabilité d’une explosion en temps fini et la loi d’une borne supérieur du temps d’explosion de la solution / This thesis is concerned with stochastic partial differential equations of parabolic type. In the first part we prove new results regarding the existence and the uniqueness of global and local variational solutions to a Neumann initial-boundary value problem for a class of non-autonomous stochastic parabolic partial differential equations. The equations we consider are defined on unbounded open domains in Euclidean space satisfying certain geometric conditions, and are driven by a multiplicative noise derived from an infinite-dimensional fractional Wiener process characterized by a sequence of Hurst parameters H = (Hi) i ∈ N+ ⊂ (1/2,1). These parameters are in fact subject to further constraints that are intimately tied up with the nature of the nonlinearity in the stochastic term of the equations, and with the choice of the functional spaces in which the problem at hand is well-posed. Our method of proof rests on compactness arguments in an essential way. The second part is devoted to the study of the blowup behavior of solutions to semilinear stochastic partial differential equations with Dirichlet boundary conditions driven by a class of differential operators including (not necessarily symmetric) Lévy processes and diffusion processes, and perturbed by a mixture of Brownian and fractional Brownian motions. Our aim is to understand the influence of the stochastic part and that of the differential operator on the blowup behavior of the solutions. In particular we derive explicit expressions for an upper and a lower bound of the blowup time of the solution and provide a sufficient condition for the existence of global positive solutions. Furthermore, we give estimates of the probability of finite time blowup and for the tail probabilities of an upper bound for the blowup time of the solutions