Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature (Q692820)

The authors study the singular limit as \(\epsilon\to 0\) of the Allen-Cahn equation \(u^{\epsilon}_t=\Delta u^{\epsilon}+\epsilon^{-2}f(u^{\epsilon})\), where \(f\) is a balanced bistable nonlinearity and initial data is independent of \(\epsilon\). They prove that the transition layer of the solutions \(u^{\epsilon}\) are sandwiched between two sharp interfaces moving by the mean curvature provided that these interfaces sandwich at \(t=0\) an \(O(\epsilon|\log \epsilon|)\) neighborhood of the initial layer. A result on the regularity of the generalized motion by mean curvature is also given.