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

Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption $$\partial_t u - \mathrm{div}(|\nabla u|^{p-2}\nabla u) +|\nabla u|^{q}=0 \qquad {\rm in} \ (0,\infty)\times\mathbb{R}^N$$ are studied when $N\geq1$ 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 $N\geq2$, existence of radially symmetric self-similar solutions is proved and a fine description of their behavior as $|x|\to\infty$ is provided.