Second order asymptotics and uniqueness for self-similar profiles to a singular diffusion equation with gradient absorption

Abstract

Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption $${\partial }_{t}u-\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)+|\nabla u{|}^q=0\mathrm{in}(0,\infty )\times {\mathbb{R}}^N$$ are studied when $N\ge 1$ and the exponents $(p,q)$ satisfy $p_c=\frac{2N}{N+1}$, $p-1<q<\frac{p}{2}$. Existence and uniqueness of such a solution are established in dimension $N=1$. In dimension $N\ge 2$, existence of radially symmetric self-similar solutions is proved and a fine description of their behavior as $|x|\to \infty$ is provided.