  For a large class of non-negative initial data, the solutions to the
quasilinear viscous Hamilton-Jacobi equation $\partial\_t u-\Delta\_p u+|\nabla
u|^q=0$ in $(0,\infty)\times\real^N$ are known to vanish identically after a
finite time when $2N/(N+1) \textless{} p \leq 2$ and $q\in(0,p-1)$. Further
properties of this extinction phenomenon are established herein:
\emph{instantaneous shrinking} of the support is shown to take place if the
initial condition $u\_0$ decays sufficiently rapidly as $|x|\to\infty$, that
is, for each $t \textgreater{} 0$, the positivity set of $u(t)$ is a bounded
subset of $\real^N$ even if $u\_0 \textgreater{} 0$ in $\real^N$. This decay
condition on $u\_0$ is also shown to be optimal by proving that the positivity
set of any solution emanating from a positive initial condition decaying at a
slower rate as $|x|\to\infty$ is the whole $\real^N$ for all times. The time
evolution of the positivity set is also studied: on the one hand, it is
included in a fixed ball for all times if it is initially bounded
(\emph{localization}). On the other hand, it converges to a single point at the
extinction time for a class of radially symmetric initial data, a phenomenon
referred to as \emph{single point extinction}. This behavior is in sharp
contrast with what happens when $q$ ranges in $[p-1,p/2)$ and $p\in
(2N/(N+1),2]$ for which we show \emph{complete extinction}. Instantaneous
shrinking and single point extinction take place in particular for the
semilinear viscous Hamilton-Jacobi equation when $p=2$ and $q\in (0,1)$ and
seem to have remained unnoticed.
