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# Nonlinear parabolic equations with a lower order term and $L^1$ data

• 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, $\gamma=\frac{(N+2)(p-1)}{N+p},$ $c(x,t)\in(L^{\tau }(Q_{T}))^N,$ $\tau=\frac{N+p}{p-1},$ $\ f\in L^1 (Q_T),$ $u_{0}\in L^1(\Omega).$

Mathematics Subject Classification: Primary: 35K60; Secondary: 35D05.

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