Well-posedness of the Vlasov-Poisson system with floating potential at a wall

We are interested in a bi-species, uni-dimensional plasma occupying the domain [0,1]. Particles are injected at x=0 and at x=1 there is a fully absorbing wall.  If we want to model the problem by the Vlasov-Poisson system, then we have to choose a boundary condition for the potential.
The system is compatible with  the Maxwell-Ampère equation only for a nonlinear Dirichlet boundary condition at the wall, which depends on currents.  This expression for the floating potential is naturally associated with finite energy solutions. We will study well-posedness of the system in the space BV.
Then, we will show simulation for stationary data, and comment on the long time behavior.