Existence of Traveling Domain Solutions for a Two-Dimensional Moving Boundary Problem

In this paper we prove the existence of a traveling domain solution for a two-dimensional moving boundary problem. Specifically, we prove the existence of a domain that travels to the right at a constant speed k and a function b which solves a porous medium type equation in the domain with constant Dirichlet boundary condition. The proof is by Schaefer’s fixed point theorem. The result may be viewed as an extension of the existence of traveling cell solutions of a one-dimensional cell motility model proved by the authors and Juliet Lee (2004).