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 $L^{1}(\Omega)$.

Keywords

• Musielak-Orlicz space
• Nonlinear Obstacle-parabolic problems
• Entropy solution