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We solve the linear advection-diffusion equation with a variable speed on asemi-infinite line. The variable speed is determined by an additional conditionat the boundary, which models the dynamics of a contact line of a hydrodynamicflow at a 180 contact angle. We use Laplace transform in spatial coordinate andGreen's function for the fourth-order diffusion equation to show localexistence of solutions of the initial-value problem associated with the set ofover-determining boundary conditions. We also analyze the explicit solution inthe case of a constant speed (dropping the additional boundary condition).

On solutions of the reduced model for the dynamical evolution of contact lines