Existence of entropy solutions in Musielak Orlicz spaces via a sequence of penalized equations

Mhamed Elmassoudi; Ahmed Aberqi; Jaouad Bennouna

doi:10.5269/bspm.v38i6.37269

Mhamed Elmassoudi University of Fez Faculty of Sciences Dhar El Mahraz Department of Mathematics Laboratory LAMA
Ahmed Aberqi University of Fez National School of Applied Sciences Fez Faculty of Sciences Dhar El Mahraz
Jaouad Bennouna University of Fez Faculty of Sciences Dhar El Mahraz Department of Mathematics Laboratory LAMA

DOI: https://doi.org/10.5269/bspm.v38i6.37269

Resumo

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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