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For a large class of non-negative initial data, the solutions to thequasilinear viscous Hamilton-Jacobi equation $\partial\_t u-\Delta\_p u+|\nablau|^q=0$ in $(0,\infty)\times\real^N$ are known to vanish identically after afinite time when $2N/(N+1) \textless{} p \leq 2$ and $q\in(0,p-1)$. Furtherproperties of this extinction phenomenon are established herein:\emph{instantaneous shrinking} of the support is shown to take place if theinitial condition $u\_0$ decays sufficiently rapidly as $|x|\to\infty$, thatis, for each $t \textgreater{} 0$, the positivity set of $u(t)$ is a boundedsubset of $\real^N$ even if $u\_0 \textgreater{} 0$ in $\real^N$. This decaycondition on $u\_0$ is also shown to be optimal by proving that the positivityset of any solution emanating from a positive initial condition decaying at aslower rate as $|x|\to\infty$ is the whole $\real^N$ for all times. The timeevolution of the positivity set is also studied: on the one hand, it isincluded 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 theextinction time for a class of radially symmetric initial data, a phenomenonreferred to as \emph{single point extinction}. This behavior is in sharpcontrast 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}. Instantaneousshrinking and single point extinction take place in particular for thesemilinear viscous Hamilton-Jacobi equation when $p=2$ and $q\in (0,1)$ andseem to have remained unnoticed.

Instantaneous shrinking and single point extinction for viscous Hamilton-Jacobi equations with fast diffusion