Nearly extremal non-trivial cross t-intersecting families and r-wise t-intersecting families

Let $n$, $r$, $k_1,\dots ,k_r$ and $t$ be positive integers with $r\ge 2$, and $F_i(1\le i\le r)$ a family of $k_i$-subsets of an $n$-set $V$. The families $F_1,F_2,\dots ,F_r$ are said to be $r$-cross $t$-intersecting if $|F_1\cap F_2\cap \cdots \cap F_r|\ge t$ for all $F_i\in F_i(1\le i\le r),$ and said to be non-trivial if $\left|{\cap }_{1\le i\le r}{\cap }_{F\in F_i}F\right|<t$. If the $r$-cross $t$-intersecting families $F_1,\dots ,F_r$ satisfy $F_1=\cdots =F_r=F$, then $F$ is well known as $r$-wise $t$-intersecting. In this paper, we first describe the structure of maximal 2-cross $t$-intersecting families with given $t$-covering numbers and then determine the structure of non-trivial 2-cross $t$-intersecting families with maximum product of their sizes. We also give a stability result for the non-trivial $r$-wise $t$-intersecting families with maximum size for $r\ge 3$.