  We consider a quasilinear parabolic differential equation associated with the
renormalization group transformation of the two-dimensional hierarchical
Coulomb system in the limit as the size of the block L goes to 1. We show that
the initial value problem is well defined in a suitable function space and the
solution converges, as t goes to infinity, to one of the countably infinite
equilibrium solutions. The nontrivial equilibrium solution bifurcates from the
trivial one. These solutions are fully described and we provide a complete
analysis of their local and global stability for all values of inverse
temperature. Gallavotti and Nicolo's conjecture on infinite sequence of
``phases transitions'' is also addressed. Our results rule out an intermediate
phase between the plasma and the Kosterlitz-Thouless phases, at least in the
hierarchical model we consider.
