## 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

APA

Chicago

ICMJE

MLA

NLM

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/10.1111/1467-9590.00074Witelski, 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.

## 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