Ionized Gas in an Annular Region

We consider a plasma that is created by a high voltage difference $\lambda$, which is known as a Townsend discharge. We consider it to be confined to the region $\Omega$ between two concentric spheres, two concentric cylinders, or more generally between two star-shaped surfaces. We first prove that if the plasma is initially relatively dilute, then either it may remain dilute for all time or it may not, depending on a certain parameter $\kappa(\lambda, \Omega)$. Secondly, we prove that there is a connected one-parameter family of steady states. This family connects the non-ionized gas to a plasma, either with a sparking voltage $\lambda^*$ or with very high ionization, at least in the cylindrical or spherical cases.