Advances in Differential Equations

Study of self-similarity for the fast-diffusion equation

Raúl Ferreira and Juan Luis Vázquez

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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

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

First available in Project Euclid: 19 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Ferreira, Raúl; Vázquez, Juan Luis. Study of self-similarity for the fast-diffusion equation. Adv. Differential Equations 8 (2003), no. 9, 1125--1152.

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