Positivity, decay, and extinction for a singular diffusion equation with gradient absorption

Details Positivity, decay, and extinction for a singular diffusion equation with gradient absorption Positivity, decay, and extinction for a singular diffusion equation with gradient absorption

2011-04-08

arxiv.org/abs/1104.1513v1

Journal of Functional Analysis 262 (2012) 3186-3239

Mathematics

Analysis of PDEs

Scientific paper

10.1016/j.jfa.2012.01.013

We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \partial_t u -\Delta_{p}u+|\nabla u|^{q}=0\quad in\;\; (0,\infty)\times\RR^N, where $N\ge 1$, $p\in(1,2)$, and $q>0$. Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as $t\to\infty$ for $q>p-N/(N+1)$, optimal decay estimates as $t\to\infty$ for $p/2\le q\le p-N/(N+1)$, or extinction in finite time for $0 < q < p/2$. In addition, we show how the diffusion prevents extinction in finite time in some ranges of exponents where extinction occurs for the non-diffusive Hamilton-Jacobi equation.

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