Cauchy problem for fast diffusion equations with a gradient term

We study the Cauchy problem for fast diffusion equation with a gradient term of the form ut−Δum=|Duq|σ,ut−Δum=|Duq|σ, where 0000 and 0<σ<20<σ<2. The existence of solutions and the optimality of the stipulated assumptions to the existence of solutions were obtained by Andreucci for the case m≥1m≥1 and qσ≥mqσ≥m. Here we extend these results to the case 00q,σ>0. However, as qσ>mqσ>m, the optimality of the stipulated assumptions is left open.