Solução Exata e Estabilização Exponencial para a Equação de Allen-Cahn

ABSTRACT

In this work we study some qualitative properties of the Allen-Cahn equation. This equation has been studied widely in several areas of science and mainly in the evolution of micro-structures during the solidification process of a pure metal or metallic alloy. The main results achieved are: the exact solution, the energy of Ginzburg-Landau and the exponential decay property of the total energy of the model. The exact solution of the problem was built from variables separation method thanks to a particular choice of the coefficient of reaction. In respect to the exponential stabilization of the total energy of solutions, we use the multiplicative techniques in order to establish the energy dissipation law and in the following we use classical inequalities of the mathematical analysis to build the estimate of exponential decay.

KEYWORDS:
diffusion equation; Allen-Cahn equation; exact solution; exponential stabilization