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

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.