Self-similar asymptotics for linear and nonlinear diffusion equations

Journal Article (Journal Article)

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.

Full Text

Duke Authors

Cited Authors

  • Witelski, TP; Bernoff, AJ

Published Date

  • January 1, 1998

Published In

Volume / Issue

  • 100 / 2

Start / End Page

  • 153 - 193

International Standard Serial Number (ISSN)

  • 0022-2526

Digital Object Identifier (DOI)

  • 10.1111/1467-9590.00074

Citation Source

  • Scopus