Let $G=(V,E)$ be a finite undirected graph. A set $S$ of vertices in $V$ is
said to be total $k$-dominating if every vertex in $V$ is adjacent to at least
$k$ vertices in $S$. The total $k$-domination number, $\gamma_{kt}(G)$, is the
minimum cardinality of a total $k$-dominating set in $G$. In this work we study
the total $k$-domination number of Cartesian product of two complete graphs
which is a lower bound of the total $k$-domination number of Cartesian product
of two graphs. We obtain new lower and upper bounds for the total
$k$-domination number of Cartesian product of two complete graphs. Some
asymptotic behaviors are obtained as a consequence of the bounds we found. In
particular, we obtain that
$\displaystyle\liminf_{n\to\infty}\frac{\gamma_{kt}(G\Box H)}{n}\leq
2\,\left(\left\lceil\frac{k}{2}\right\rceil^{-1}+\left\lfloor\frac{k+4}{2}\right\rfloor^{-1}\right)^{-1}$
for graphs $G,H$ with order at least $n$. We also prove that the equality is
attained if and only if $k$ is even. The equality holds when $G,H$ are both
isomorphic to the complete graph, $K_n$, with $n$ vertices. Furthermore, we
obtain closed formulas for the total $2$-domination number of Cartesian product
of two complete graphs of whatever order. Besides, we prove that, for $k=3$,
the inequality above is improvable to $\displaystyle\liminf_{n\to\infty}
\gamma_{3t}(K_n\Box K_n)/n \leq 11/5$.
