Sharp Lr asymptotics of the small solutions to the nonlinear Schrödinger equations

Abstract

This paper studies the large time behavior of the small solution to the nonlinear Schrödinger equation with power type nonlinearity. If the power is large enough, then it is well known that the nonlinear solution asymptotically behaves like a linear solution as t→±∞ (see e.g. (J. Funct. Anal. 32 (1979) 1; J. Math. Pures Appl. 64 (1985) 363)). Our concern at the present work is to determine the sharp decay rate of the difference between the nonlinear solution and the linear solution in L r (2⩽r⩽∞) spaces.