Solução exata e estabilização exponencial para a equação de Allen-Cahn - TEIXEIRA, Edson Danilo da Paixão (2022)
In this paper we study some quantitative and qualitative properties of the Allen-Cahn equation.The Allen-Cahn equation has been widely studied in several areas of science and especially in the evolution of microstructures during the solidification process of a pure metal or alloy. The main results obtained in this work are: the exact solution, the Ginzburg-Landau energy G(t) and the exponential decay property of the system.The analytical resolution of the problem was obtained by the variable separation method thanks to an appropriate choice of the reaction coefficient. With this, we start to consider a problem of initial value and another of contour value, which are solved. Regarding the exponential stabilization of total energy, we use multiplicative techniques to find the dissipation law and from there we apply the Poincaré and Jensen inequalities to construct the exponential decay estimate.