Summary: Let \(G = (V, E)\) be a graph of no isolated vertex, a real function \(f : E(G)\to [0, 1]\) is said to be a fractional star domination function (FSDF) of \(G\) if \(\sum\limits_{uv\in E} f(uv) \geq 1\) holds for every vertex \(u \in V(G)\), the fractional star domination number \(\gamma^\prime_{fs}(G)\) of \(G\) is defined as \(\gamma^\prime_{fs}(G) = \min\{\sum\limits_{uv\in E} f(uv)\}\) (\(f\) is an FSDF of \(G\)). In this paper, we discuss some questions on the fractional star domination, give the fractional star domination numbers of some special graphs, and determine the fractional star domination numbers for the product graphs of \(P_m \times P_n\) and \(C_m \times P_n\).