A Qualitative Research on Allen-Cahn Equations

Abstract

Originated in the study of the evolution of non-conserved order fields during anti-phase domain coarsening, the Allen-Cahn equation has the form [special characters omitted] Numerous applications of Allen-Cahn equations are found in fields relating to phase transition. In this dissertation, the Allen-Cahn equation with one-dimensional spatial variable is considered and solutions connecting two stable states are studied. A symmetry and classification result will be given for both balanced and unbalanced double-well potentials.