In this paper we prove the existence of a renormalized solution for a class of
nonlinear parabolic problems whose prototype is
$\frac{\partial u}{\partial t}-\Delta_p u+$ div $(c(x,t)|u|^{\gamma-1}u) =f $ in $Q_T$
$u(x,t)=0$ on $\partial\Omega\times(0,T) $
$u(x,0)=u_0 (x)$ in $\Omega,$
where $Q_T=\Omega\times(0,T),$ $\Omega$ is an open and
bounded subset of $
\mathcal{R}
^N$, $N\geq2,$ $T>0,$ $\Delta_p$ is the so called $p$-Laplace operator,
$\gamma=\frac{(N+2)(p-1)}{N+p},$ $c(x,t)\in(L^{\tau
}(Q_{T}))^N,$ $\tau=\frac{N+p}{p-1},$ $\ f\in L^1 (Q_T), $
$u_{0}\in L^1(\Omega).$
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