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Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2016.tde-08032016-101021
Document
Author
Full name
Salvador Rafael Manjate
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2015
Supervisor
Committee
Pereira, Marcone Corrêa (President)
Aragão, Gleiciane da Silva
Marrocos, Marcus Antonio Mendonça
Title in Portuguese
Um problema semilinear elíptico em domínio fino com termos de reações concentradas na fronteira
Keywords in Portuguese
Domínios finos
Equações semilineares elípticas
Equações singulares
Semicontinuidade superior e inferior
Termos concentrados
Abstract in Portuguese
Analisamos o comportamento assintótico de uma família de equilíbrios de uma equação de reação-difusão com a condição de Neumann homegênea definido num domínio fino bidimensional com termos de reação concentradas em uma vizinhança oscilante da fronteira. Assumimos que o domínio e portanto a vizinhança oscilante se degeneram em um intervalo quando o parâmetro positivo tende a zero. O objetivo principal foi mostrar que essa família de soluções, converge para uma equação limite unidimensional, que captura a geometria e o comportamento assintótico dos conjuntos abertos onde o problema é estabelecido. De fato mostramos a continuidade da família de equilíbrios
Title in English
Semilinear elliptic equations in thin domains with reaction terms concentrating on boundary
Keywords in English
Concentrating terms
Lower semicontinuity
Semilinear elliptic equations
Singular elliptic equations
Thin domains
Upper semicontinuity
Abstract in English
In this work we analyze the behavior of a family of stead state solutions of a semilinear reaction-diffusion equation with homogeneous Neumann boundary condition, posed in a two-dimensional thin domain with reaction term concentrated in an narrow oscillating neighborhood. Indeed, we assume that the domain of definition of the solutions degenerates into an interval as a small parameter $\epsilon$ goes to zero. Our main result is that this family of solutions converge to the solution of a one-dimensional limit equation capturing the geometry and oscillatory behaviour of the open sets where the problem is estabilished.
 
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Publishing Date
2016-09-12
 
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