Large time behavior for a quasilinear diffusion equation with critical gradient absorption

International audienceWe study the large time behavior of non-negative solutions to thenonlinear diffusion equation with critical gradient absorption\partial_t u-\Delta_{p}u+|\nabla u|^{q_*}=0 \quad \hbox{in} \(0,\infty)\times\mathbb{R}^N\ ,for $p\in(2,\infty)$ and $q_*:=p-N/(N+1)$. We show that theasymptotic profile of compactly supported solutions is given by asource-type self-similar solution of the $p$-Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results

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

International audienceFor 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 $(0,\infty)\times\mathbb{R}^N$ are known to vanish identically after a finite time when $2N/(N+1) 0$, the positivity set of $u(t)$ is a bounded subset of $\mathbb{R}^N$ even if $u_0 > 0$ in $\mathbb{R}^N$. 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 $\mathbb{R}^N$ 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