  By combining the formalism of \cite{RHE} with a discrete approach close to
the considerations of \cite{Davie}, we interpret and solve the rough partial
differential equation $dy_t=A y_t \, dt+\sum_{i=1}^m f_i(y_t) \, dx^i_t$ ($t\in
[0,T]$) on a compact domain $\mathcal{O}$ of $\R^n$, where $A$ is a rather
general elliptic operator of $L^p(\mathcal{O})$ ($p>1$),
$f_i(\vp)(\xi):=f_i(\vp(\xi))$ and $x$ is the generator of a $2$-rough path.
The (global) existence, uniqueness and continuity of a solution is established
under classical regularity assumptions for $f_i$. Some identification
procedures are also provided in order to justify our interpretation of the
problem.
