Evolution of a stable profile for a class of nonlinear diffusion equations. II

Abstract

First, explicit formulas are found for all the eigenfunctions and eigenvalues of a Sturm–Liouville problem associated with the class of nonlinear diffusion equations studied previously. The formulas for the eigenfunctions are proportional to Gegenbauer polynomials whose argument depends on the separable solution shape function. Next, rigorous bounds on the asymptotic amplitude are found in terms of integrals of the initial data. These bounds are the best possible bounds of the given type since they produce the exact result for the separable solution. Finally, results of numerical experiments are reported for D∼nδ where δ=1, −1/3, −1/2, and −2/3. The rigorous bounds are compared to the perturbation estimates from the earlier work and to the computed values of the asymptotic amplitude.