I know that the problem \begin{equation} \Delta_p u = f \end{equation} make sense if $f \in L^q$ with $n/p<q<n$ and that is there a existence theory for $$ u_t -\Delta_p u = f $$ For a given open and bounded set $U \subset \mathbb{R}^{n}$ and $T>0$ we shall denote the parabolic cylinder $U_{T} = U \times (0,T]$. For the inhomogeneity term $f: U_{T} \rightarrow \mathbb{R}$ we shall consider $$ f \in L^{q,r}(U_{T}) := L^{r}(0,T, L^{q}(U)) $$ if $$ 2/r+n/q>1. $$ What are the analogs for these problem if we are working with the problems $$ u_t -\Delta_{G^p} u = f $$ with $$ f \in L^{G^q,r}(U_{T}) := L^{G^q}(0,T, L^{q}(U))$$ $G^q$ and $G^p$ is an N-function.
In other words, how to generalize this p-Laplacian problem for an Orlicz problem?
