Analysis Seminar

---

The Tale of Two Tails

Speaker: Philip ROSENAU, Tel-Aviv University

Location: Warren Weaver Hall 1302

Date: Wednesday, October 24, 2018, 11 a.m.

Synopsis:

We discuss formation of patterns due to Fisher-KPP reaction appended with fast or slow diffusions
\begin{equation}
u_{t} = \left [ D(u) u_{x} \right ]_{x} + u(1 - � u)
\end{equation}
where
\begin{equation}
D(u) = \left \{
\begin{array}{cl}
u, & \text{slow diffusion} \\
1, & \text{Standard Fisher-KPP}\\
\frac{1}{u}, & \text{fast (logarithmic)}
\end{array}
\right .
\end{equation}
In the fast diffusion case the problem of travelling waves, TW, is mapped into a linear problem with the propagation speed $\lambda$ being selected via a boundary condition(s), b.c., imposed at the far away upstream. In the Dirichlet case the process relaxes into a steady state, whereas convective b.c.; $u_{x}+hu = 0$, lead the system into a heating (cooling)TW for $h < 1$ ($1 < h$) and into an equilibrium
if $h = 1$. We derive {\it explicit solutions} of both expanding and collapsing formations that quench within a finite time. The later  being a unique feature of 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 expanding formations are strong attractors of their respective initial excitations.