Solução Exata e Estabilização Exponencial para a Equação de Allen-Cahn
DOI:
https://doi.org/10.5540/tcam.2021.022.04.00559Keywords:
Equação de difusão, Equação de Allen-Cahn, Solução exata, Estabilização exponencial.Abstract
Neste trabalho estudamos algumas propriedades qualitativas da equação de Allen-Cahn. Esta equação tem sido amplamente estudada em diversas áreas da ciência e principalmente na evolução de microestruturas durante o processo de solidicação de um metal puro ou liga metálica. Os principais resultados obtidos neste trabalho são a solução exata, a energia de Ginzburg-Landau e a propriedade de decaimento exponencial do sistema. A solução analítica do problema foi obtida pelo método da separação de variáveis graças a uma escolha adequada do coeciente de reação. Com isto, passamos a considerar dois problemas, um problema de valor inicial e o outro de valor de contorno, em que ambos foram resolvidos. Em relação a estabilização exponencial da energia total de solução, usamos técnicas multiplicativas para estabelecer a lei de dissipação da energia e, em seguida, algumas desigualdades para construir a estimativa de decaimento exponencial.
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