A coupled PDE/ODE model of mitochondrial swelling: Large-time behavior of the homogeneous Dirichlet problem

In this paper, we are continuing our analysis of a coupled PDE/ODE model of calcium induced mitochondria swelling. We study the long term behavior under Dirichlet boundary conditions, for which the analytical machinery that has been developed previously for Neumann conditions does not apply and must be extended. It is shown that in this setting the calcium ion concentration will tend to zero and that in general complete swelling will not take place as t → ∞. This phenomenon, which is due to an implicitly enforced calcium ion flux across the boundary, distinguishes the situation under Dirichlet conditions from the situation under Neumann conditions that were analyzed previously. This is a preliminary study in which the analytical arguments are developed that will be required lateron to study a more realistic model of the biological system, where Robin and/or mixed boundary conditions must be considered.