Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

Abstract

Abstract. We consider the derivative nonlinear Schrödinger equationswhere the coefficient a (t) satisfies the time growth conditionis a sufficiently small constant and the nonlinear interaction term F consists of cubic nonlinearities of derivative typewhere λ 1 , λ 6 ∈ R, λ 2 , λ 3 , λ 4 , λ 5 ∈ C, λ 2 − λ 3 ∈ R, and λ 4 − λ 5 ∈ R. We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate u(t) L p ≤ C t