RESUMEN

We investigate similarity solutions of the "thin film" equation. In particular we look at solutions on the half-line $x\ge0$ with compact support and zero contact angle boundary conditions in x=0. Such "dipole" solutions feature an anomalous exponent and are therefore called similarity solutions of the second kind. Using a combination of phase space analysis and numerical simulations, we numerically construct trajectories representing these solutions, at the same time obtaining broader insight into the nature of the four-dimensional phase space. Additional asymptotic analysis provides further information concerning the evolution to self-similarity.

RESUMEN

We address the small-time evolution of interfaces (fronts) for the pseudoparabolic  generalization δu/δt=δ/δx (uα δu/δx + uβ δ2u/δxδt) of the porous-medium equation, identifying regimes in which the local behavior remains fixed for some finite time and others in which it changes instantaneously. A number of phenomena beyond those exhibited by the porous-medium equation are elucidated, including retreating fronts and novel types of local behavior. Related results for the important limit case  δu/δt=δ/δx (uβ δ2u/δxδt)