An entropy solution for some degenerate or singular obstacle parabolic problems with $L^1-$data via a sequence of penalized equations

Abstract

We give an existence result of the obstacle parabolic degenerate or singular problem associated to the equation, $\displaystyle\frac{\partial u}{\partial t} + A(u) = f$ in $Q_T,$ where $A$ is a classical Leray-Lions operator acting from the weighted Sobolev space $L^p(0,T,W_0^{1,p}(\Omega,w))$ into its dual $L^{p'}(0,T,W^{-1,p'}(\Omega,w^*))$, while the datum $f$ is assumed to lie in $L^1(Q_T)$. The proof is based on the penalization methods.