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 u t − Δ u m = | D u q | σ , where 0 < m < 1 , q > 0 and 0 < 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 ≥ 1 and q σ ≥ m . Here we extend these results to the case 0 < m < 1 and all q , σ > 0 . However, as q σ > m , the optimality of the stipulated assumptions is left open.