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August  2020, 40(8): 4801-4819. doi: 10.3934/dcds.2020202

Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation

1. 

Sorbonne Université & CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France

2. 

Department of Mathematics, Hangzhou Normal University, 2318 Yuhangtang Road, 311121 Hangzhou, China

* Corresponding author

Received  October 2019 Published  May 2020

Fund Project: ZH thanks NSFC 11671353, 11401153, Zhejiang Provincial Natural Science Foundation of China under Grant No. LY18A010025, and CSC for their financial support; and the Laboratoire JacquesLouis Lions for its kind hospitality

We study the time-asymptotic behavior of solutions of the Schrödinger equation with nonlinear dissipation
$ \begin{equation*} \partial _t u = i \Delta u + \lambda |u|^\alpha u \end{equation*} $
in
$ {\mathbb R}^N $
,
$ N\geq1 $
, where
$ \lambda\in {\mathbb C} $
,
$ \Re \lambda <0 $
and
$ 0<\alpha<\frac2N $
. We give a precise description of the behavior of the solutions (including decay rates in
$ L^2 $
and
$ L^\infty $
, and asymptotic profile), for a class of arbitrarily large initial data, under the additional assumption that
$ \alpha $
is sufficiently close to
$ \frac2N $
.
Citation: Thierry Cazenave, Zheng Han. Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4801-4819. doi: 10.3934/dcds.2020202
References:
[1]

T. CazenaveS. CorreiaF. Dickstein and F. B. Weissler, A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation, São Paulo J. Math. Sci., 9 (2015), 146-161.  doi: 10.1007/s40863-015-0020-6.  Google Scholar

[2]

T. Cazenave, Z. Han and Y. Martel, Blowup on an arbitrary compact set for a Schrödinger equation with nonlinear source term, J. Dynam. Differential Equations (2020). doi: 10.1007/s10884-020-09841-8.  Google Scholar

[3]

T. CazenaveY. Martel and L. F. Zhao, Finite-time blowup for a Schrödinger equation with nonlinear source term, Discrete Contin. Dynam. Systems, 39 (2019), 1171-1183.  doi: 10.3934/dcds.2019050.  Google Scholar

[4]

T. Cazenave and I. Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Commun. Contemp. Math., 19 (2017), 1650038, 20 pp. doi: 10.1142/S0219199716500383.  Google Scholar

[5]

T. Cazenave and I. Naumkin, Modified scattering for the critical nonlinear Schrödinger equation, J. Funct. Anal., 274 (2018), 402-432.  doi: 10.1016/j.jfa.2017.10.022.  Google Scholar

[6]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.  doi: 10.1007/BF02099529.  Google Scholar

[7]

M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112.  doi: 10.1103/RevModPhys.65.851.  Google Scholar

[8]

J. GinibreT. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 211-239.   Google Scholar

[9]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅱ. Scattering theory, general case, J. Funct. Anal., 32 (1979), 33-71.  doi: 10.1016/0022-1236(79)90077-6.  Google Scholar

[10]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅲ. Special theories in dimensions 1, 2 and 3, Ann. Inst. Henri Poincaré Sect. A (N.S.), 28 (1978), 287-316.   Google Scholar

[11]

N. Hayashi, C. H. Li and P. I. Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical fields, Adv. Math. Phys., (2016), Art. ID 3702738, 7 pp. doi: 10.1155/2016/3702738.  Google Scholar

[12]

N. HayashiC. H. Li and P. I. Naumkin, Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 16 (2017), 2089-2104.  doi: 10.3934/cpaa.2017103.  Google Scholar

[13]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.   Google Scholar

[14]

T. Kato, Nonlinear Schrödinger equations, Schrödinger Operators (Sønderborg, 1988), Lecture Notes in Phys., Springer, Berlin, 345 (1989), 218-263.  doi: 10.1007/3-540-51783-9_22.  Google Scholar

[15]

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 (2009), 39-64.  doi: 10.2969/jmsj/06110039.  Google Scholar

[16]

A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 759-834.  doi: 10.1016/S1874-575X(02)80036-4.  Google Scholar

[17]

K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 45-68.  doi: 10.1007/s00030-002-8118-9.  Google Scholar

[18]

T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 25 (2006), 403-408.  doi: 10.1007/s00526-005-0349-2.  Google Scholar

[19]

A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations, 31 (2006), 1407-1423.  doi: 10.1080/03605300600910316.  Google Scholar

[20]

K. Stewartson and J. T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech., 48 (1971), 529-545.  doi: 10.1017/S0022112071001733.  Google Scholar

[21]

W. A. Strauss, Nonlinear scattering theory at low energy: Sequel, J. Funct. Anal., 43 (1981), 281-293.  doi: 10.1016/0022-1236(81)90019-7.  Google Scholar

show all references

References:
[1]

T. CazenaveS. CorreiaF. Dickstein and F. B. Weissler, A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation, São Paulo J. Math. Sci., 9 (2015), 146-161.  doi: 10.1007/s40863-015-0020-6.  Google Scholar

[2]

T. Cazenave, Z. Han and Y. Martel, Blowup on an arbitrary compact set for a Schrödinger equation with nonlinear source term, J. Dynam. Differential Equations (2020). doi: 10.1007/s10884-020-09841-8.  Google Scholar

[3]

T. CazenaveY. Martel and L. F. Zhao, Finite-time blowup for a Schrödinger equation with nonlinear source term, Discrete Contin. Dynam. Systems, 39 (2019), 1171-1183.  doi: 10.3934/dcds.2019050.  Google Scholar

[4]

T. Cazenave and I. Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Commun. Contemp. Math., 19 (2017), 1650038, 20 pp. doi: 10.1142/S0219199716500383.  Google Scholar

[5]

T. Cazenave and I. Naumkin, Modified scattering for the critical nonlinear Schrödinger equation, J. Funct. Anal., 274 (2018), 402-432.  doi: 10.1016/j.jfa.2017.10.022.  Google Scholar

[6]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.  doi: 10.1007/BF02099529.  Google Scholar

[7]

M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112.  doi: 10.1103/RevModPhys.65.851.  Google Scholar

[8]

J. GinibreT. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 211-239.   Google Scholar

[9]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅱ. Scattering theory, general case, J. Funct. Anal., 32 (1979), 33-71.  doi: 10.1016/0022-1236(79)90077-6.  Google Scholar

[10]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅲ. Special theories in dimensions 1, 2 and 3, Ann. Inst. Henri Poincaré Sect. A (N.S.), 28 (1978), 287-316.   Google Scholar

[11]

N. Hayashi, C. H. Li and P. I. Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical fields, Adv. Math. Phys., (2016), Art. ID 3702738, 7 pp. doi: 10.1155/2016/3702738.  Google Scholar

[12]

N. HayashiC. H. Li and P. I. Naumkin, Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 16 (2017), 2089-2104.  doi: 10.3934/cpaa.2017103.  Google Scholar

[13]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.   Google Scholar

[14]

T. Kato, Nonlinear Schrödinger equations, Schrödinger Operators (Sønderborg, 1988), Lecture Notes in Phys., Springer, Berlin, 345 (1989), 218-263.  doi: 10.1007/3-540-51783-9_22.  Google Scholar

[15]

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 (2009), 39-64.  doi: 10.2969/jmsj/06110039.  Google Scholar

[16]

A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 759-834.  doi: 10.1016/S1874-575X(02)80036-4.  Google Scholar

[17]

K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 45-68.  doi: 10.1007/s00030-002-8118-9.  Google Scholar

[18]

T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 25 (2006), 403-408.  doi: 10.1007/s00526-005-0349-2.  Google Scholar

[19]

A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations, 31 (2006), 1407-1423.  doi: 10.1080/03605300600910316.  Google Scholar

[20]

K. Stewartson and J. T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech., 48 (1971), 529-545.  doi: 10.1017/S0022112071001733.  Google Scholar

[21]

W. A. Strauss, Nonlinear scattering theory at low energy: Sequel, J. Funct. Anal., 43 (1981), 281-293.  doi: 10.1016/0022-1236(81)90019-7.  Google Scholar

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