We recall the definition of $\mathcal G$-quasiasymptotics at infinity in a framework of Colombeau space $\mathcal G$ (cf. [8]) and give an application of that notion to a Cauchy problem for a strictly semilinear hyperbolic system. It turns out that quasiasymptotic behaviour at infinity of the solution inherits the quasiasymptotic behaviour at infinity of initial data under suitable assumptions on the nonlinear term.