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August  2015, 35(8): 3343-3376. doi: 10.3934/dcds.2015.35.3343

On weak interaction between a ground state and a trapping potential

Scipio Cuccagna 1, and  Masaya Maeda 2,

1. 

Department of Mathematics and Geosciences, University of Trieste, via Valerio 12/1 Trieste, 34127, Italy
2. 

Department of Mathematics and Informatics, Faculty of Science, Chiba University, Chiba 263-8522, Japan

Received  April 2014 Revised  December 2014 Published  February 2015

We continue our study initiated in [4] of the interaction of a ground state with a potential considering here a class of trapping potentials. We track the precise asymptotic behavior of the solution if the interaction is weak, either because the ground state moves away from the potential or is very fast.
Keywords: ground states, Nonlinear Schrödinger equation, asymptotic stability..
Mathematics Subject Classification: Primary: 35Q5.
Citation: Scipio Cuccagna, Masaya Maeda. On weak interaction between a ground state and a trapping potential. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3343-3376. doi: 10.3934/dcds.2015.35.3343
References:
[1]

Rend. Istit. Mat. Univ. Trieste, 44 (2012), 197-257.  Google Scholar

[2]

Comm. Math. Physics, 305 (2011), 279-331. doi: 10.1007/s00220-011-1265-2.  Google Scholar

[3]

Trans. Amer. Math. Soc., 366 (2014), 2827-2888. doi: 10.1090/S0002-9947-2014-05770-X.  Google Scholar

[4]

J. Differential Equations, 256 (2014), 1395-1466. doi: 10.1016/j.jde.2013.11.002.  Google Scholar

[5]

S. Cuccagna and M. Maeda, On small energy stabilization in the NLS with a trapping potential,, preprint, ().   Google Scholar

[6]

Comm. Pure Appl. Math., 58 (2005), 1-29. doi: 10.1002/cpa.20050.  Google Scholar

[7]

Comm. Partial Differential Equations, 34 (2009), 1074-1113. doi: 10.1080/03605300903076831.  Google Scholar

[8]

Acta Math., 188 (2002), 163-262. doi: 10.1007/BF02392683.  Google Scholar

[9]

Jour. Funct. An., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[10]

Int. Math. Res. Not., 66 (2004), 3559-3584. doi: 10.1155/S1073792804132340.  Google Scholar

[11]

Comm. Math. Physics, 274 (2007), 187-216. doi: 10.1007/s00220-007-0261-z.  Google Scholar

[12]

J. Nonlinear Sci., 17 (2007), 349-367. doi: 10.1007/s00332-006-0807-9.  Google Scholar

[13]

Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 287-317. doi: 10.1017/S030821051000003X.  Google Scholar

[14]

Duke Math. J., 133 (2006), 405-466. doi: 10.1215/S0012-7094-06-13331-8.  Google Scholar

[15]

Ann. Inst. H. Poinc. Anal. Non Lin., 28 (2011), 357-384. doi: 10.1016/j.anihpc.2011.02.002.  Google Scholar

[16]

Math. Res. Lett., 16 (2009), 477-486. doi: 10.4310/MRL.2009.v16.n3.a8.  Google Scholar

[17]

Comm. Partial Diff., 29 (2004), 1051-1095. doi: 10.1081/PDE-200033754.  Google Scholar

[18]

Comm. Pure Appl. Math., 58 (2005), 149-216. doi: 10.1002/cpa.20066.  Google Scholar

[19]

I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS,, preprint, ().   Google Scholar

[20]

Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.  Google Scholar

show all references

References:
[1]

Rend. Istit. Mat. Univ. Trieste, 44 (2012), 197-257.  Google Scholar

[2]

Comm. Math. Physics, 305 (2011), 279-331. doi: 10.1007/s00220-011-1265-2.  Google Scholar

[3]

Trans. Amer. Math. Soc., 366 (2014), 2827-2888. doi: 10.1090/S0002-9947-2014-05770-X.  Google Scholar

[4]

J. Differential Equations, 256 (2014), 1395-1466. doi: 10.1016/j.jde.2013.11.002.  Google Scholar

[5]

S. Cuccagna and M. Maeda, On small energy stabilization in the NLS with a trapping potential,, preprint, ().   Google Scholar

[6]

Comm. Pure Appl. Math., 58 (2005), 1-29. doi: 10.1002/cpa.20050.  Google Scholar

[7]

Comm. Partial Differential Equations, 34 (2009), 1074-1113. doi: 10.1080/03605300903076831.  Google Scholar

[8]

Acta Math., 188 (2002), 163-262. doi: 10.1007/BF02392683.  Google Scholar

[9]

Jour. Funct. An., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[10]

Int. Math. Res. Not., 66 (2004), 3559-3584. doi: 10.1155/S1073792804132340.  Google Scholar

[11]

Comm. Math. Physics, 274 (2007), 187-216. doi: 10.1007/s00220-007-0261-z.  Google Scholar

[12]

J. Nonlinear Sci., 17 (2007), 349-367. doi: 10.1007/s00332-006-0807-9.  Google Scholar

[13]

Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 287-317. doi: 10.1017/S030821051000003X.  Google Scholar

[14]

Duke Math. J., 133 (2006), 405-466. doi: 10.1215/S0012-7094-06-13331-8.  Google Scholar

[15]

Ann. Inst. H. Poinc. Anal. Non Lin., 28 (2011), 357-384. doi: 10.1016/j.anihpc.2011.02.002.  Google Scholar

[16]

Math. Res. Lett., 16 (2009), 477-486. doi: 10.4310/MRL.2009.v16.n3.a8.  Google Scholar

[17]

Comm. Partial Diff., 29 (2004), 1051-1095. doi: 10.1081/PDE-200033754.  Google Scholar

[18]

Comm. Pure Appl. Math., 58 (2005), 149-216. doi: 10.1002/cpa.20066.  Google Scholar

[19]

I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS,, preprint, ().   Google Scholar

[20]

Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.  Google Scholar

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