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Doctoral Thesis
DOI
10.11606/T.55.2013.tde-06112013-165332
Document
Author
Full name
Rawlilson de Oliveira Araujo
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2013
Supervisor
Committee
Fu, Ma To (President)
Calsavara, Bianca Morelli Rodolfo
Federson, Márcia Cristina Anderson Braz
Menzala, Gustavo Alberto Perla
Sare, Hugo Danilo Fernandez
Title in Portuguese
Estabilidade assintótica de uma classe de equações quasilineares viscoelásticas com história
Keywords in Portuguese
Atrator global
Atratores exponenciais
Equação da onda
Equações diferencias parciais
Memória
Unicidade
Viscoelasticidade
Abstract in Portuguese
Este trabalho é dedicado ao estudo do comportamento a longo prazo de uma classe de equações viscoelásticas não lineares com memória, da forma |'upsilon IND. t'| POT. ho' 'upsilon IND. tt' - DELTA 'upsilon' - 'DELTA upsilon IND. tt' + 'INT. SUP. t INF. \tau' upsilon (t- s) 'DELTA epsilon' (s) ds = h, '\tau' > 0, definida num domínio limitado de 'R POT. N'. Tal classe de problemas foi estudada por diversos autores desde 2001, com '\tau = 0. Os resultados existentes são principalmente devotados à existência de soluções globais, decaimento da energia, com ou sem dissipações adicionais, existência com dados pequenos, entre outros. Entretanto, a questão da unicidade de soluções e existência de atratores globais não foram discutidas em trabalhos anteriores. No presente trabalho, apresentamos resultados de unicidade e existência de atratores globais para essa classe de problemas num contexto mais geral, incluindo o caso em que '\tau' = -'INFINITO'. Além disso, incluímos um problema complementar, de quarta ordem onde estudamos a existência de atratores exponenciais
Title in English
Asymptotic stability for a class of quasilinear viscoelastic equations with past history
Keywords in English
Exponential attractors
Global attractor
Memory
Partial differential equations
Uniqueness
Viscoelasticity
Wave equation
Abstract in English
This work is concerned with the long-time behaviour of a class nonlinear viscoelastic equations of the form |'upsilon IND. t'| POT. ho' 'upsilon IND. tt' - DELTA 'upsilon' - 'DELTA upsilon IND. tt' + 'INT. SUP. t INF. \tau' upsilon (t- s) 'DELTA epsilon' (s) ds = h, 'ho' > 0, defined in a bounded domain of 'R POT. N'. Such class of problems was studied by several authors since 2001, with '\tau' = 0. Existing results are mainly devoted to global existence, energy decay, with or without additional dampings, existence with small data, among others. However, uniqueness and existence of global attractors were not considered previously. In the present work, we establish some results on the uniqueness of solutions and existence of global attractors in a more general setting, including '\tau' = - 'INFINITY'. In addition, we have added a second problem concerned with a fourth order equation where we study the existence of exponential attractors
 
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Publishing Date
2013-11-08
 
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