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

Abstract

We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \partial_t u -Δ_{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.

References in corpus (2)

• Localized non-diffusive asymptotic patterns for nonlinear parabolic equations with gradient absorption
• Gradient estimates for a degenerate parabolic equation with gradient absorption and applications

Cited by in corpus (3)

• Global and blow-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids
• Eternal solutions to a singular diffusion equation with critical gradient absorption
• Asymptotic behavior for a singular diffusion equation with gradient absorption