2003 Study of self-similarity for the fast-diffusion equation
Raúl Ferreira, Juan Luis Vázquez
Adv. Differential Equations 8(9): 1125-1152 (2003). DOI: 10.57262/ade/1355926582


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.


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Raúl Ferreira. Juan Luis Vázquez. "Study of self-similarity for the fast-diffusion equation." Adv. Differential Equations 8 (9) 1125 - 1152, 2003. https://doi.org/10.57262/ade/1355926582


Published: 2003
First available in Project Euclid: 19 December 2012

zbMATH: 1032.35106
MathSciNet: MR1989292
Digital Object Identifier: 10.57262/ade/1355926582

Primary: 35K57
Secondary: 34C11 , 34D05

Rights: Copyright © 2003 Khayyam Publishing, Inc.


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Vol.8 • No. 9 • 2003
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