Well-posedness of a Debye type system endowed with a full wave equation

We prove well-posedness for a transport-diffusion problem coupled with a wave equation for the potential. We assume that the initial data are small. A bilinear form in the spirit of Kato’s proof for the Navier–Stokes equations is used, coupled with suitable estimates in Chemin–Lerner spaces. In the one dimensional case, we get well-posedness for arbitrarily large initial data by using Gagliardo–Nirenberg inequalities.