Study of self-similarity for the fast-diffusion equation (Q1413043)

The article is devoted to studying the existence and properties of special classes of solutions for nonlinear evolution equation \[ u_t = (u^{m-1}u_x)_x, \quad x\in \mathbb{R},\;t \geq 0 \] in the range of parameters \(m\leq 0\) (very fast diffusion). The authors investigate the existence and properties of travelling waves and of self-similar solutions of three types: forward in time, backward in time and exponential type. The problem is reduced to solving an ordinary differential equations. The main aim is to perform a hopefully complete study of the ordinary differential equation obtained with a view to classifying the types of behavior that appear, mark the difference with \(m > 0\), and derive suitable consequences for the general theory of the original problem. In particular, there exists three options in the behaviour at infinity or near a singularity (extending the standards choice between slow and fast rates). The novelty is the existence of solutions with very fast decay as \(|x|\to\infty\).