Self-similar Asymptotics for Linear and Nonlinear Diffusion Equations
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.
© 1998 Massachusetts Institute of Technology
Witelski, T. P. and Bernoff, A. J. (1998), Self-similar Asymptotics for Linear and Nonlinear Diffusion Equations. Studies in Applied Mathematics, 100: 153–193. doi: 10.1111/1467-9590.00074