Asymptotic behavior for a singular diffusion equation with gradient absorption

We study the large time behavior of non-negative solutions to the singular diffusion equation with gradient absorption ∂ t u − Δ p u + | ∇ u | q = 0 in ( 0 , ∞ ) × R N , for p c : = 2 N / ( N + 1 ) p 2 and p / 2 q q ⁎ : = p − N / ( N + 1 ) . We prove that there exists a unique very singular solution of the equation, which has self-similar form and we show the convergence of general solutions with suitable initial data towards this unique very singular solution.