Study of the Equilibria of Parabolic Differential Equations with Interfaces Intersecting the Boundary

Author

Michal Kowalczyk, University of Tennessee - Knoxville

Date of Award

8-1995

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Nicholas Alikakos

Committee Members

Don Hinton, A. Freire, H. Simpson, S. Georghiou

Abstract

Existence of steady state solutions for the Allen-Cahn and Cahn-Hilliard equations in two dimensional domains is discussed. We are in particular interested in establishing existence of single layered equilibria with the property that their transition layer intersects the boundary. In the case of the Allen-Cahn equation we consider bone-like domains and seek solutions intersecting the flat part of the boundary. We establish conditions for the domain which ensure existence of such equilibria. Their stability is also analyzed. For the Cahn-Hilliard equations we show that there exist equilibria near every point of a local maximum of the curvature of the boundary.

Recommended Citation

Kowalczyk, Michal, "Study of the Equilibria of Parabolic Differential Equations with Interfaces Intersecting the Boundary. " PhD diss., University of Tennessee, 1995.
https://trace.tennessee.edu/utk_graddiss/3534