Existence results for a Cauchy–Dirichlet parabolic problem with a repulsive gradient term

We study the existence of solutions of a nonlinear parabolic problem of Cauchy-Dirichlet type having a lower order term which depends on the gradient. The model we have in mind is the following: $$ u_t-div (A(t,x)nabla u|nabla u|^{p-2})=gamma |Nabla u|^q+f(t,x) quad in quad Q_T, $$ $$ u=0 quad on quad(0,T) imes partial Omega, $$ u(0,x)=u_0(x) quad in quad Omega, $$ where $Q_T=(0,T) imes Omega$, $Omega$ is a bounded domain of $mathrm{R}^N$, $Nge 2$, $1