Note on the 3-dimensional log canonical abundance in characteristic $>3$

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 $\kappa (K_X+B)\ge 0$, and if $K_X+B$ is nef and $\kappa (K_X+B)\ge 1$, 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 $n(K_X+B)\le 2$ or when the Albanese map $a_X:X\to \mathrm{A}\mathrm{l}\mathrm{b}(X)$ is non-trivial. Moreover, we prove that the abundance for klt threefold pairs over $k$ implies the abundance for log canonical threefold pairs over $k$.