The purpose of this paper is to verify the existence, uniqueness, and asymptotic behaviours of global strong solutions of the exterior nonstationary problem for the Navier-Stokes equation in \(L_ q\) spaces. The \(L_ q-L_ r\) estimates for the semigroup \(e^{-tA}\) generated by the Stokes operator in an exterior domain \(\Omega \subset {\mathbb{R}}^ n\), \(n\geq 3\), plays crucial roles and this paper is essentially devoted to the study of the \(L_ q-L_ r\) estimates. We show that \(\| e^{- tA}f\|_{L_ r(\Omega)}\leq Ct^{-\sigma}\| f\|_{L_ q(\Omega)},\) \(t>0\), where \(1