The coupled system of the Ohm-Maxwell and transport equations is solved applying a 1-D code to the case of a one-fluid plasma in a cylindrical Tokamak. A situation is considered such that the current channel is surrounded by a large and cold boundary region with high thermal conductivity simulating the effect of strong cooling processes in the boundary zone. Introducing suitable boundary conditions on the interface a resistive instability exists (in addition to the thermal instability related to local overheating) which is related to marginal modes of the equation for the perturbed poloidal flux (plasma-circuit instability). The current shrinks and collapses on the axis on the time scale depending on the resistivity of the cold zone and on its width. If tau E approximately na2R the nonlinear evolution of the profiles is controlled by the parameter nqRU/B (U is the loop voltage) and a very steep gradient is developed at high density.