Existence of entropy solutions in Musielak Orlicz spaces via a sequence of penalized equations
Abstract
This paper, is devoted to an existence result of entropy unilateral solutions for the nonlinear parabolic problems with obstacle in Musielak- Orlicz--spaces:
$$ \partial_{t}u + A(u) + H(x,t,u,\nabla u) =f + div(\Phi(x,t,u))$$
and $$ u\geq \zeta \,\,\mbox{a.e. in }\,\,Q_T.$$
Where $A$ is a pseudomonotone operator of Leray-Lions type defined in the inhomogeneous Musielak-Orlicz space $W_{0}^{1,x}L_{\varphi}(Q_{T})$,
$H(x,t,s,\xi)$ and $\phi(x,t,s)$ are only assumed to be Crath\'eodory's functions satisfying only the growth conditions prescribed by Musielak-Orlicz functions $\varphi$ and $\psi$ which inhomogeneous and does not satisfies $\Delta_2$-condition. The data $f$ and $u_{0}$ are still taken in $L^{1}(Q_T)$ and $L^{1}(\Omega)$.
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