LAUSR

(0.1) { i∂tu+ 1 2∆u = λ |u| 2 n u, (t, x) ∈ R × R, u (0, x) = u0 (x) , x ∈ R in space dimensions n ≥ 4, where λ ∈ R. In the case of 1 ≤ n ≤ 3, asymptotic behavior of small amplitude solutions to (0.1) has been studied in [1], [2], [3], [4], [5]. However in the case of n ≥ 4, there are no results for asymptotic behavior of solutions as far as we know. Our purpose in this talk is to show the sharp asymptotics and time decay of solutions to (0.1) in the uniform norm for higher space dimensions n ≥ 4. We introduce some function spaces and notations. Let L∞ ∩ C denote the bounded continuous function space with the norm ∥φ∥L∞∩C = sup x∈Rn |φ (x)| . The