We study the competition between the Rayleigh-Taylor instability and stabilization by the vertical flow through the boundary ofphase transition in geothermal systems in dependence on the thickness of the low-permeability layer and permeability for fixed pressure drop and homogeneous conditions at the instant borders of the low-permeability layer. This competition becomes apparent by non-uniqueness of localization ofplane phase transition fronts, at least one of which being always unstable. The multiple fronts appear as a result of bifurcation from the unique stable front. In a neighborhood of the bifurcation point dynamics of a narrow band of weakly unstable and weakly nonlinear modes is described by one form of the nonlinear Kolmogorov-Petrovsky-Piscounov equation (KPP).