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.