Electric breakdown and ionization fronts are considered theoretically in a sandwich-like dc discharge system consisting of two plane-parallel electrodes and a gaseous gap in between. The key system feature is a high-ohmic cathode opposite to an ordinary metal anode. Such systems have received much attention from experimental studies because they naturally support current patterns. Using adiabatic description of electrons and two-scale expansion we demonstrate that in the low-current Townsend mode the discharge is governed by a two-component reaction-diffusion system. The latter provides quantitative system description on the macroscopic time scale (i.e., much larger than the ion travel time). The breakdown appears as an instability of the uniform overvoltage state. A seed current fluctuation triggers a shock-like ionization front that propagates along the discharge plane with constant speed (typically approximately 10(4) cm/s). Depending on the cathode resistivity the front exhibits either monotonic or oscillatory behavior in space. Other breakdown features, such as damping transient oscillations of the global current, can also be found as solutions of the reaction-diffusion equations.