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.

No description available.

Thesis (MScIng)--University of Stellenbosch, 2006. / Bibliography. Also available via the Internet.

Note:

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.

Orientador: Rafael Zadorosny / Resumo: Os materiais supercondutores são extremamente promissores a novas aplicações. Podemos encontrar diversos estudos sobre o supercondutor YBCO, porém o tratamento térmico para a formação da fase YBa2Cu3O7-δ (Y123) na forma de nanofibras ainda não é bem definido na literatura. Para estudar de forma sistemática essa questão, e definir uma boa rota de tratamento térmico, fios com diâmetros em torno de 300 nm foram obtidos por Solution Blow Spinning (SBS) a partir de uma solução contendo acetatos metálicos de ítrio, bário e cobre, polímero PVP (polivinilpirrolidona), ácidos acético e propiônico, e metanol. Os fios poliméricos foram submetidos a um tratamento térmico para obtenção da fase Y123. Este tratamento foi dividido em duas partes, a primeira variando a temperatura máxima de sinterização (850°C, 875°C, 900°C e 925°C) por 1 hora. E a segunda variando o tempo em que as fibras ficaram submetidas à temperatura de 925°C (10 min, 30 min, 1h, 3h e 6h). Para analisar a influência da temperatura e do tempo de sinterização foram utilizados Microscopia Eletrônica de Varredura (MEV), Difração de raios-X (DRX) e Magnetometria AC. As análises de DRX indicaram a presença do YBa2Cu3O7-δ em todas as amostras, porém, com fases secundárias. As medidas de susceptibilidade magnética revelaram influência da temperatura de sinterização na blindagem magnética da amostra, indicando que as amostras tratadas a 925°C/1h obtiveram a melhor resposta magnética. Sendo assim, o melhor tratamento térmico foi o... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: Although the production of YBa2Cu3O7-δ (Y123) has been extensively reported in the literature, there is still a lack of information concerning the ideal heat-treatment routes applied to Y123 nanowires. Thus, wires with diameters around 300 nm were prepared by Solution Blow Spinning (SBS) from a solution containing yttrium, barium and copper metal acetates, PVP polymer (polyvinylpyrrolidone), acetic and propionic acids, and methanol. The resulting polymer wires were heat-treated to obtain the Y123 ceramic phase. Those treatments were divided into two parts: in the first one, which lasted 1 hour, the maximum sintering temperature was changed (850°C, 875°C, 900°C, and 925°C); in the second one, the time of the plateau at 925°C was altered (10 min, 30 min, 1h, 3h, and 6h). The structural characterization was carried out by Scanning Electron Microscopy (SEM), X-ray Diffraction (XRD), and AC Magnetometry. SEM images revealed that all samples yielded wires with an average diameter ranging from 241 nm to 376 nm. XRD analysis indicated the presence of the secondary phase BaCuO2 in all samples, as well as the phase Y123. Magnetic measurements corroborate the other characterizations, and indicate the sample sintered at 925 °C/1h as the one giving the highest magnetic response. / Mestre

In this study, one of the plastic automotive parts, Air Duct, manufactured through blow moulding process is reviewed and investigated with a practical process point of view using structural mechanics approach. First, current blow moulding process was examined to find governing factors of the process which can be improved or adjusted for better quality control of finished product. Secondly, numerical analysis was conducted on the post-mould process in order to predict the deformation in the final products with properly assumed initial and boundary conditions. The simulation results showed that the degree of warpage under current blow moulding process could be predicted at a reasonable accuracy. It was also discovered that the distortions of the holes are strongly dependent on it location and surrounding, and the current cooling method should be improved to improve the quality. Based on the simulation results and literature survey, a better post-mould cooling method was suggested. In addition, the problem in cooling system was identified, and redesigning scheme was recommended.

In this study, one of the plastic automotive parts, Air Duct, manufactured through blow moulding process is reviewed and investigated with a practical process point of view using structural mechanics approach. First, current blow moulding process was examined to find governing factors of the process which can be improved or adjusted for better quality control of finished product. Secondly, numerical analysis was conducted on the post-mould process in order to predict the deformation in the final products with properly assumed initial and boundary conditions. The simulation results showed that the degree of warpage under current blow moulding process could be predicted at a reasonable accuracy. It was also discovered that the distortions of the holes are strongly dependent on it location and surrounding, and the current cooling method should be improved to improve the quality. Based on the simulation results and literature survey, a better post-mould cooling method was suggested. In addition, the problem in cooling system was identified, and redesigning scheme was recommended.