  In this paper, we prove the non-vanishing and some special cases of the
abundance for log canonical threefold pairs over an algebraically closed field
$k$ of characteristic $p > 3$. More precisely, we prove that if $(X,B)$ be a
projective log canonical threefold pair over $k$ and $K_{X}+B$ is
pseudo-effective, then $\kappa(K_{X}+B)\geq 0$, and if $K_{X}+B$ is nef and
$\kappa(K_{X}+B)\geq 1$, then $K_{X}+B$ is semi-ample.
  As applications, we show that the log canonical rings of projective log
canonical threefold pairs over $k$ are finitely generated and the abundance
holds when the nef dimension $n(K_{X}+B)\leq 2$ or when the Albanese map
$a_{X}:X\to \mathrm{Alb}(X)$ is non-trivial. Moreover, we prove that the
abundance for klt threefold pairs over $k$ implies the abundance for log
canonical threefold pairs over $k$.
