The authors deal with a two-scale microstructure model of current flow in a medium with continuously distributed capacitance. The model is extended to include nonlinearities in the conductance across the interface between the local capacitors and the global conducting medium. The operator, describing the model, has the abstact form of a system of degenerate semilinear parabolic equations. The operator \(A\) that arises from a distributed capacitance model is the subgradient of a convex function, i.e., there exists a convex function \(\phi\) such that \(A= \partial\phi\). Then there exists a convex function \(\Phi\) such that the equation of the model is equivalent to an explicit equation of the standard form \(g \in ({d}/{dt}) w + \partial\Phi (w)\), so this problem is parabolic and has the corresponding regularizing effects on the data. Various limiting cases are identified and the corresponding convergence results are obtained by letting selected parameters tend to infinity.