This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., . Our aim is to obtain the two results. One asserts that, if the -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of in .
J.E.Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phy.25(11) (1984), 3270–3273. doi:10.1063/1.526074.
2.
T.Cazenave and I.Naumkin, Modified scattering for the critical nonlinear Schrödinger equation, J. Funct. Anal.274(2) (2018), 402–432. doi:10.1016/j.jfa.2017.10.022.
3.
N.Hayashi, C.Li and P.I.Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical fields, Adv. Math. Phys. Art.2016 (2016), Article ID 3702738, 7 pp.
4.
N.Hayashi and P.I.Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math.120(2) (1998), 369–389. doi:10.1353/ajm.1998.0011.
5.
N.Kita, Optimal decay rate of solutions to 1D Schrödinger equation with cubic dissipative nonlinearity, J. Applied Science and Engineering A.1(1) (2019), 15–18.
6.
N.Kita and A.Shimomura, Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data, J. Math. Soc. Japan.61(1) (2009), 39–64. doi:10.2969/jmsj/06110039.
7.
T.Ogawa and T.Sato, -Decay rate for the critical nonlinear Schrödinger equation with a small smooth data, NoDEA Nonlinear Differential Equations Appl.27(2) (2020), Paper No. 18, 20 pp.
8.
T.Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys.139(3) (1991), 479–493. doi:10.1007/BF02101876.
9.
T.Sato, -Decay estimate for the dissipative nonlinear Schrödinger equation in the Gevrey class, Arch. Math. (Basel)115(5) (2020), 575–588. doi:10.1007/s00013-020-01483-y.
10.
A.Shimomura, Asymptotic behavior of solutions to Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations.31(7–9) (2006), 1407–1423. doi:10.1080/03605300600910316.
11.
H.Sunagawa, Large time behavior of solutions to the Klein–Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan.58(2) (2006), 379–400. doi:10.2969/jmsj/1149166781.
