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 formut−div(|Du|p−2Du)=|Duqσ|ut−div(|Du|p−2Du)=|Duq|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 qσ>p−1qσ>p−1, and obtains the existence of solutions on the assumptions that measures as initial data for the case 0p−1qσ>p−1, the optimality of the stipulated assumptions to existence of solutions is still open.