  We study the dynamics of a one-dimensional non-linear and non-local drift-di
usion equation set in the half-line, with the coupling involving the trace
value on the boundary. The initial mass M of the density determines the
behaviour of the equation: attraction to self similar pro le, to a steady state
of nite time blow up for supercritical mass. Using the logarithmic Sobolev and
the HWI inequalities we obtain a rate of convergence for the cases subcritical
and critical mass. Moreover, we prove a comparison principle on the equation
obtained after space integration. This concentration-comparison principle
allows proving blow-up of solutions for large initial data without any
monotonicity assumption on the initial data.
