On global solutions to the initial-boundary value problem for the damped nonlinear Schrödinger equations

Abstract

where a > 0 and Q is a bounded domain in RN with smooth boundary X?. When a = 0 (undamped case), the global existence of smooth solutions to the initial-boundary value problem (l)-(3) has been proved only when N=2 and f(lu12)w lulp with p = 2 (Brezis and Gallouet Cl]), with 2 I p I3 (M. Tsutsumi [9]). Higher dimensional cases are very difficult pribl&s and remain unproved as far as the author knows. If we add a linear damping term iau (a > 0), then the mass Ilull iZtR1 and the energy { IVu(x, t)l* dx + j F( lu(x, t)l*) dx decay exponentially as t + cc (as is seen below), which may assert the global existence of (small) smooth solutions even if the nonlinear term f(lul’)u does not have a proper sign. It is our purpose to establish the global existence and decay estimates of smooth solutions of the problem (l)-(3) with small data. There are a number of works on the nonlinear evolution equations (see [3, 5-71 and their references) in the following spirit: “Energy and decay estimates give global existence of solutions.” If the damping term is not added, there is no hope that the decay estimates of solutions in some F(Q) (p 2 2) can be obtained since