  We solve 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. We use Laplace transform in spatial coordinate and
Green's function for the fourth-order diffusion equation to show local
existence of solutions of the initial-value problem associated with the set of
over-determining boundary conditions. We also analyze the explicit solution in
the case of a constant speed (dropping the additional boundary condition).
