Nonlinear parabolic systems in Musielack-Orlicz space

Abstract

In this paper, we discuss the solvability of the nonlinear parabolic systems associated to the nonlinear parabolic equation: for $i=1,2$
\begin{equation*}
\mathcal{(S)} \left\{\begin{array}{lll}
\displaystyle\frac{\partial u_{i}}{\partial t} -div(a(x,t,u_{i},\nabla u_{i}))+ g_{i}(x,t,u_{i},\nabla u_i)) =f_{i}(x,u_{1},u_{2})\, \quad & \mbox{in}\quad & Q_{T},\\
\displaystyle u_{i}(x,t)=0 &\mbox{on}& \partial\Omega \times(0,T),\\
\displaystyle u_{i}(x,(t=0))=u_{i,0}(x) & \mbox{in} & \Omega,
\end{array}%
\right.
\end{equation*}
with the source $f$ is merely integrable. The operator $\displaystyle A(u)= div \Big (a(x,t,u_{i},\nabla u_{i})\Big)$
is a generalized Leray-Lions operator defined on the inhomogeneous Musielak-Orlicz spaces (the vector field $\displaystyle a(x,t,u_i,\nabla u_i)$ have a growth prescribed by a generalized N-function). The non linearity $g_{i}$ is a Carath\'{e}odory function satisfy the some condition.