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Self-similar asymptotics for linear and nonlinear diffusion equations

Publication ,  Journal Article
Witelski, TP; Bernoff, AJ
Published in: Studies in Applied Mathematics
January 1, 1998

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.

Duke Scholars

Published In

Studies in Applied Mathematics

DOI

ISSN

0022-2526

Publication Date

January 1, 1998

Volume

100

Issue

2

Start / End Page

153 / 193

Related Subject Headings

  • Mathematical Physics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics
 

Citation

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Witelski, T. P., & Bernoff, A. J. (1998). Self-similar asymptotics for linear and nonlinear diffusion equations. Studies in Applied Mathematics, 100(2), 153–193. https://doi.org/10.1111/1467-9590.00074
Witelski, T. P., and A. J. Bernoff. “Self-similar asymptotics for linear and nonlinear diffusion equations.” Studies in Applied Mathematics 100, no. 2 (January 1, 1998): 153–93. https://doi.org/10.1111/1467-9590.00074.
Witelski TP, Bernoff AJ. Self-similar asymptotics for linear and nonlinear diffusion equations. Studies in Applied Mathematics. 1998 Jan 1;100(2):153–93.
Witelski, T. P., and A. J. Bernoff. “Self-similar asymptotics for linear and nonlinear diffusion equations.” Studies in Applied Mathematics, vol. 100, no. 2, Jan. 1998, pp. 153–93. Scopus, doi:10.1111/1467-9590.00074.
Witelski TP, Bernoff AJ. Self-similar asymptotics for linear and nonlinear diffusion equations. Studies in Applied Mathematics. 1998 Jan 1;100(2):153–193.
Journal cover image

Published In

Studies in Applied Mathematics

DOI

ISSN

0022-2526

Publication Date

January 1, 1998

Volume

100

Issue

2

Start / End Page

153 / 193

Related Subject Headings

  • Mathematical Physics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics