Evolution of a stable profile for a class of nonlinear diffusion equations. III. Slow diffusion on the line

Abstract

We show that the nonlinear diffusion equation ∂n/∂t =∂2(n1+δ/∂x2 with compact initial data on −∞<x<∞ can be transformed into another nonlinear diffusion equation α(∂ϑ∂t)=ϑ1+α ×∂2ϑ/∂?2 on a fixed finite interval of the ? axis. Thus, the original moving boundary problem is transformed into a fixed boundary problem. The new form of the equation has advantages both analytically and computationally as the examples illustrate. Linear stability analysis for the transformed equation is straightforward whereas the moving boundaries of the original problem complicate the analysis for that case. The advantages of the resulting computational algorithm for solving the moving boundary problem are also discussed. A nonlinear Rayleigh–Ritz quotient and a Lyapunov functional are shown to be bounded, monotonically decreasing functions of time. Both functionals give added insight into the mathematical character of the diffusion process.