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July  2010, 9(4): 929-942. doi: 10.3934/cpaa.2010.9.929

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

1. 

Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli "Federico II", Complesso Monte S.Angelo, via Cintia, 80126 Napoli, Italy

Received  May 2009 Revised  January 2010 Published  April 2010

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), $ $u_{0}\in L^1(\Omega).$

Citation: Rosaria Di Nardo. Nonlinear parabolic equations with a lower order term and $L^1$ data. Communications on Pure & Applied Analysis, 2010, 9 (4) : 929-942. doi: 10.3934/cpaa.2010.9.929
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