On $r$-cross $t$-intersecting families for weak compositions

Let $\mathbb N_0$ be the set of non-negative integers, and let $P(n,l)$ denote the set of all weak compositions of $n$ with $l$ parts, i.e., $P(n,l)=\{ (x_1,x_2,\dots, x_l)\in\mathbb N_0^l\ :\ x_1+x_2+\cdots+x_l=n\}$. For any element $\mathbf u=(u_1,u_2,\dots, u_l)\in P(n,l)$, denote its $i$th-coordinate by $\mathbf u(i)$, i.e., $\mathbf u(i)=u_i$. Let $l=\min(l_1,l_2,\dots, l_r)$. Families $\mathcal A_j\subseteq P(n_j,l_j)$ ($j=1,2,\dots, r$) are said to be $r$-cross $t$-intersecting if $\vert \{ i\in [l] \ :\ \mathbf u_1(i)=\mathbf u_2(i)=\cdots=\mathbf u_r(i)\} \vert\geq t$ for all $\mathbf u_j\in \mathcal A_j$. Suppose that $l\geq t+2$. We prove that there exists a constant $n_0=n_0(l_1,l_2,\dots,l_r,t)$ depending only on $l_j$'s and $t$, such that for all $n_j\geq n_0$, if the families $\mathcal A_j\subseteq P(n_j,l_j)$ ($j=1,2,\dots, r$) are $r$-cross $t$-intersecting, then \begin{equation} \prod_{j=1}^r \vert \mathcal{A}_j \vert\leq \prod_{j=1}^r {n_j+l_j-t-1 \choose l_j-t-1}.\notag \end{equation} Moreover, equality holds if and only if there is a $t$-set $T$ of $\{1,2,\dots,l\}$ such that $\mathcal{A}_j=\{\mathbf u\in P(n_j,l_j)\ :\ \mathbf u(i)=0\ {\rm for\ all}\ i\in T\}$ for $j=1,2,\dots, r$.