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

TL;DR

This paper investigates the extinction phenomena of solutions to viscous Hamilton-Jacobi equations, revealing conditions for instantaneous support shrinking and single point extinction, especially for specific parameter ranges and initial data decay rates.

Contribution

It establishes the occurrence of instantaneous shrinking and single point extinction phenomena, identifying optimal decay conditions and contrasting behaviors across parameter ranges.

Findings

Support shrinks instantaneously if initial data decays rapidly.

Positivity set remains bounded if initially bounded.

Solutions can converge to a single point at extinction time.

Abstract

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.