Blow-up of solutions of the semilinear equations

Author(s)

Fernanda Tomé Alves

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2006-03-31

Examining board members

Alexandre Nolasco de Carvalho; Olimpio Hiroshi Miyagaki; Sérgio Henrique Monari Soares

• Advisor: Alexandre Nolasco de Carvalho

Abstract

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. (AU)