Self‐similar Asymptotics for Linear and Nonlinear Diffusion Equations

Abstract

The long‐time asymptotic solutions of initial value problems for the heat equation and the nonlinear porous medium equation are self‐similar spreading solutions. The symmetries of the governing equations yield three‐parameter families of these solutions given in terms of their mass, center of mass, and variance. Unlike the mass and center of mass, the variance, or “time‐shift,” of a solution is not a conserved quantity for the nonlinear problem. We derive an optimal linear estimate of the long‐time variance. Newman's Lyapunov functional is used to produce a maximum entropy time‐shift estimate. Results are applied to nonlinear merging and time‐dependent, inhomogeneously forced diffusion problems.