  We study finite-time singularities in the linear advection-diffusion equation
with a variable speed on a semi-infinite line. The variable speed is determined
by an additional condition at the boundary, which models the dynamics of a
contact line of a hydrodynamic flow at a 180 contact angle. Using apriori
energy estimates, we derive conditions on variable speed that guarantee that a
sufficiently smooth solution of the linear advection--diffusion equation blows
up in a finite time. Using the class of self-similar solutions to the linear
advection-diffusion equation, we find the blow-up rate of singularity
formation. This blow-up rate does not agree with previous numerical simulations
of the model problem.
