Existence of solutions for degenerate parabolic equations with singular terms

In this paper we deal with parabolic problems whose simplest model is $$\{\begin{array}{ll}{u}^{\mathrm{\prime }}-{\mathrm{\Delta }}_{p}u+B\frac{\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p}{u}=0& \text{in}(0,T)\times \mathrm{\Omega },\\ u(0,x)=u_0(x)& \text{in }\mathrm{\Omega },\\ u(t,x)=0& \text{on }(0,T)\times \mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $T>0$, $N\ge 2$, $p>1$, $B>0$, and $u_0$ is a positive function in $L^{\mathrm{\infty }}(\mathrm{\Omega })$ bounded away from zero.