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

Razvan Gabriel Iagar (1, 2) , Philippe Laurencot (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IMAR - "Simion Stoilow" Institute of Mathematics

We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \begin{equation*} \partial_t u -\Delta_{p}u+|\nabla u|^{q}=0\quad \mbox{ in }\;\; (0,\infty)\times\RR^N, \end{equation*} 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.

Mots clés: Singular diffusion; gradient absorption; gradient estimates; extinction; p-Laplacian; viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]

Razvan Gabriel Iagar, Philippe Laurencot. Positivity, decay, and extinction for a singular diffusion equation with gradient absorption. Journal of Functional Analysis, 2012, 262, pp.3186-3239. ⟨10.1016/j.jfa.2012.01.013⟩. ⟨hal-00584302⟩