| Contents |
| We consider the non\-linear evolutionary problem
for an unknown function $u = u(x,t)$,
%
\begin{equation}
\label{e:ivp}
\left.
\begin{aligned}
& \partial_t u =
\partial_x\left( |\partial_x u|^{p-2} \partial_x u \right) + f(u) \,,
\quad x\in \mathbb{R}^1 \,,\ t > 0 \,;
\\
& u(\,\cdot\,,0) = u_0 \quad\mbox{ in }\, \mathbb{R}^1 \,.
\end{aligned}
\qquad\right\}
\end{equation}
%
Here,
$1 < p < \infty$,
$f\colon \mathbb{R}\to \mathbb{R}$
is of Fisher\--Kolmogorov\--type, and
$u_0\colon \mathbb{R}^1\to \mathbb{R}$ are given data
between the two extremal zeros $0$ and $1$ of $f$.
%
Under some ``natural'' hypotheses on the nonlinearities,
we first derive the existence and uniqueness of
a {\it travelling wave\/}
(valued in $(0,1)$ or $[0,1]$),
then discuss applications to
$\;$ (i)
the existence and uniqueness of weak (semigroup) solutions and
$\;$ (ii)
the existence and stability of monotone travelling waves connecting
the equilibrium states $0$ and $1$.
The monotone travelling waves connecting
the equilibrium states $0$ and $1$ can
either only approach these equilibria at $\pm \infty$,
or else attain them at finite points, depending on
the interaction between the degenerate / singular diffusion and
the nonsmooth reaction function.
Then we discuss the approach to such travelling waves by solutions with
rather general initial data that are sqeezed between
two travelling waves (that are each other's shift). |
|