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 ∂ t u − div ( | ∇ u | p − 2 ∇ u ) + | ∇ u | q = 0 in ( 0 , ∞ ) × R N are studied when N ⩾ 1 and the exponents ( p , q ) satisfy p c = 2 N N + 1 p 2 , p − 1 q p 2 . Existence and uniqueness of such a solution are established in dimension N = 1. In dimension N ⩾ 2 , existence of radially symmetric self-similar solutions is proved and a fine description of their behaviour as | x | → ∞ is provided.