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Master's Dissertation
DOI
10.11606/D.55.2006.tde-23022007-103210
Document
Author
Full name
Fernanda Tomé Alves
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2006
Supervisor
Committee
Carvalho, Alexandre Nolasco de (President)
Miyagaki, Olimpio Hiroshi
Soares, Sérgio Henrique Monari
Title in Portuguese
Blow-up de soluções positivas de equações semilineares
Keywords in Portuguese
Blow-up
Equações semilineares
Abstract in Portuguese
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
Title in English
Blow-up of solutions of the semilinear equations
Keywords in English
Blow-up
Semilinear equations
Abstract in English
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.
 
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Publishing Date
2007-02-28
 
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