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 $${\mathrm{\partial }}_{t}u-{\mathrm{\Delta }}_{p}u+\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q=0\text{in}(0,\mathrm{\infty })\times {\mathrm{\Re }}^N,$$ for $p_c:=2N/(N+1)