# The Tale of Two Tails

Abtract

We discuss formation of patterns due to Fisher-KPP reaction appended with fast or slow diffusions

$u_{t} = \left [ D(u) u_{x} \right ]_{x} + u(1 - u)$

where

$D(u) = \left \{\begin{array}{cl} u, & \text{slow diffusion} \\ 1, & \text{Standard Fisher-KPP}\\ \frac{1}{u}, & \text{fast (logarithmic)} \end{array} \right . $

In the *Fast Diffusion* case the problem of travelling waves, TW, is mapped into a linear problem with the propagation speed $\lambda$ being selected by the far away boundary condition(s). Imposing the natural convective b.c.; $u_{x}+hu = 0$, leads the system into a heating (cooling)TW for $h < 1$ ($1 < h$) and if $h = 1$ the system relaxes into an equilibrium. We derive e*xplicit solutions* of both expanding and *collapsing formations that quench within a finite time*. The later being a unique feature of the fast diffusion.

In the Slow Diffusion case wherein $D(u) = u$, unfolding a hidden symmetry we map the problem into *it a purely diffusive process* and thus demonstrate that both the semi-compact Travelling Kinks and certain expanding formations are strong attractors of their respective initial excitations.

Curiously enough, though slow and fast processes are completely different processes, one can map both problems into new variables which reveal new families of explicit solutions used in their analysis.