## Advances in Differential Equations

### Study of self-similarity for the fast-diffusion equation

#### Abstract

We classify special solutions $u>0$ for the diffusion equation $$u_t=\left(u^{m-1} u_x\right)_{x},$$ in the very singular range of parameters, $m\le 0$ (very fast diffusion). We investigate the existence and properties of traveling waves and of self-similar solutions of three types: forward in time, backward in time and exponential type. The study is a necessary preliminary for the construction of a general existence and uniqueness theory of very fast diffusion. We find interesting differences with respect to the standard range $m>0$. In particular, there exist three options in the behaviour at infinity or near a singularity (extending the standard choice between slow and fast rates). The novelty is the existence of solutions with very fast decay as $|x|\to \infty$. There are other ways in which this range differs from usual nonlinear diffusion. Thus, we construct very singular solutions and show that there exists not only one as in the case $m>0$, but infinitely many; moreover, they are classical solution.

#### Article information

Source
Adv. Differential Equations Volume 8, Number 9 (2003), 1125-1152.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355926582

Mathematical Reviews number (MathSciNet)
MR1989292

Zentralblatt MATH identifier
1032.35106

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 34C11: Growth, boundedness 34D05: Asymptotic properties

#### Citation

Ferreira, Raúl; Vázquez, Juan Luis. Study of self-similarity for the fast-diffusion equation. Adv. Differential Equations 8 (2003), no. 9, 1125--1152. https://projecteuclid.org/euclid.ade/1355926582.