Consider the $N$-component coupled linear heat equation in an open subset of $\mathbb{R}\times\mathbb{R}^d$: $\forall \alpha \in \{1,...,N\}$ $$\partial_t u_{\alpha} = \sum_{j,k=1}^d\sum_{\beta=1}^N \partial_j[D^{j\alpha k\beta}(x,t) \partial_ku_{\beta}]$$ where $D$ is positive definite (viewing the index pair $j\alpha$ as the row of the matrix and $k\beta$ as the column.
Q: Is there a natural extension of the maximum theorem of the $N=1$ case to the case of higher $N$?