Asymptotic behavior of the nonlinear damped Schrödinger equation

Inui, Takahisa

doi:10.1090/proc/14276

Asymptotic behavior of the nonlinear damped Schrödinger equation
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by Takahisa Inui PDF

Proc. Amer. Math. Soc. 147 (2019), 763-773 Request permission

Abstract:

We are interested in the asymptotic behavior of the solution to the nonlinear damped Schrödinger equation (NLDS). In the present paper, we discuss when the global solutions to NLDS exponentially scatter to linear damped solutions. Moreover, we also show the additional time decay order.

References

Jacqueline E. Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phys. 25 (1984), no. 11, 3270–3273. MR 761850, DOI 10.1063/1.526074
Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
Guanggan Chen, Jian Zhang, and Yunyun Wei, A small initial data criterion of global existence for the damped nonlinear Schrödinger equation, J. Phys. A 42 (2009), no. 5, 055205, 7. MR 2525379, DOI 10.1088/1751-8113/42/5/055205
Thierry Cazenave and Fred B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992), no. 1, 75–100. MR 1171761
G. Fibich, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math. 61 (2001), no. 5, 1680–1705. MR 1824520, DOI 10.1137/S0036139999362609
Martin V. Goldman, K. Rypdal, and B. Hafizi, Dimensionality and dissipation in Langmuir collapse, Phys. Fluids 23 (1980), no. 5, 945–955. MR 573378, DOI 10.1063/1.863074
Nakao Hayashi and Pavel I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math. 120 (1998), no. 2, 369–389. MR 1613646
Tosio Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113–129 (English, with French summary). MR 877998
Darwich Mohamad, Blow-up for the damped $L^2$-critical nonlinear Schrödinger equation, Adv. Differential Equations 17 (2012), no. 3-4, 337–367. MR 2919105
Masahito Ohta and Grozdena Todorova, Remarks on global existence and blowup for damped nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst. 23 (2009), no. 4, 1313–1325. MR 2461853, DOI 10.3934/dcds.2009.23.1313
Masayoshi Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, SIAM J. Math. Anal. 15 (1984), no. 2, 357–366. MR 731873, DOI 10.1137/0515028
Masayoshi Tsutsumi, On global solutions to the initial-boundary value problem for the damped nonlinear Schrödinger equations, J. Math. Anal. Appl. 145 (1990), no. 2, 328–341. MR 1038160, DOI 10.1016/0022-247X(90)90403-3