On the Cauchy problem for the singular parabolic equations with gradient term

The Cauchy problem for a singular parabolic equation with gradient term of the form u t − div ( | D u | p − 2 D u ) = | D u q | σ is studied. This paper both extends the existence of solutions of the porous media equation with the same gradient term to this Cauchy problem for the case q σ > p − 1 , and obtains the existence of solutions on the assumptions that measures as initial data for the case 0 < q σ ⩽ p − 1 . However, as q σ > p − 1 , the optimality of the stipulated assumptions to existence of solutions is still open.