Patric,
It would be helpful to get details on the PDE formulation that shows how the domain evolves. I assume domain refers to geometry otherwise my comments that follow are not relevant. PDE Toolbox (for 2-D spatial problems) works directly with fixed geometry. Evolving geometries can sometimes be mapped to this fixed geometry via a transformation. Example: Lagrangian formulation for geometrically nonlinear problems. The general idea is to change the PDE (constitutive equations, Boundary conditions) such that the geometry is fixed.
Best, Deepak