Nonlinear diffusion: Self-similarity and traveling-waves

The presence of fluids in earth materials causes the propagation of fractures, which is observed on large scales to be regulated by nonlinear diffusion processes. In contrast with linear diffusion processes, nonlinear diffusion can result in flows with a pulse-like character (i.e., the fluid is confined to a finite space) whose profiles remain unchanged in time and which possess either self-similar or traveling-wave characters. Entropy-like (Lyapunov functional) arguments are developed here to demonstrate the asymptotic trend to self-similarity or traveling-waves in a broad class of nonlinear diffusion problems.