Existence of front solutions for a nonlocal transport problem describing gas ionization

We discuss a moving boundary problem arising from a model of gas ionization in the case of negligible electron diffusion and suitable initial data. It describes the time evolution of an ionization front. Mathematically, it can be considered as a system of transport equations with different characteristics for positive and negative charge densities. We show that only advancing fronts are possible and prove short-time well-posedness of the problem in Hölder spaces of functions. Technically, the proof is based on a fixed point argument for a Volterra type system of integral equations involving potential operators. It crucially relies on estimates of such operators with respect to variable domains in weighted Hölder spaces and related calculus estimates.