Persistence property in weighted Sobolev spaces for nonlinear dispersive equations
Authors:
X. Carvajal and W. Neves
Journal:
Quart. Appl. Math. 73 (2015), 493-510
MSC (2010):
Primary 35A01, 35Q53
DOI:
https://doi.org/10.1090/qam/1387
Published electronically:
June 12, 2015
MathSciNet review:
3400755
Full-text PDF Free Access
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Abstract: We generalize the Abstract Interpolation Lemma proved by the authors in Carvajal and Neves (2010). Using this extension, we show in a more general context the persistence property for the generalized Korteweg-de Vries equation in the weighted Sobolev space with low regularity in the weight. The method used can be applied for other nonlinear dispersive models, for instance the multidimensional nonlinear Schrödinger equation.
References
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
- X. Carvajal and W. Neves, Persistence of solutions to higher order nonlinear Schrödinger equation, J. Differential Equations 249 (2010), no. 9, 2214–2236. MR 2718656, DOI https://doi.org/10.1016/j.jde.2010.05.013
- Nakao Hayashi, Kuniaki Nakamitsu, and Masayoshi Tsutsumi, Nonlinear Schrödinger equations in weighted Sobolev spaces, Funkcial. Ekvac. 31 (1988), no. 3, 363–381. MR 987792
- Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR 759907
- Tosio Kato and Kyūya Masuda, Nonlinear evolution equations and analyticity. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 6, 455–467 (English, with French summary). MR 870865
- Joules Nahas, A decay property of solutions to the k-generalized KdV equation, Adv. Differential Equations 17 (2012), no. 9-10, 833–858. MR 2985676
- J. Nahas, A decay property of solutions to the mKdV equation, PhD thesis, University of California-Santa Barbara, June 2010.
- J. Nahas and G. Ponce, On the persistence properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces, Harmonic analysis and nonlinear partial differential equations, RIMS Kôkyûroku Bessatsu, B26, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 23–36. MR 2883844
- J. Nahas and G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation, Comm. Partial Differential Equations 34 (2009), no. 10-12, 1208–1227. MR 2581970, DOI https://doi.org/10.1080/03605300903129044
- T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), no. 2, 201–222. MR 1741843
- K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, Coupled higher-order nonlinear Schrödinger equations in nonlinear optics: Painlevé analysis and integrability, Phys. Rev. E (3) 50 (1994), no. 2, 1543–1547. MR 1381874, DOI https://doi.org/10.1103/PhysRevE.50.1543
- Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. MR 1696311
- Hideo Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations 4 (1999), no. 4, 561–580. MR 1693278
- Yoshio Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115–125. MR 915266
References
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin, 1976. MR 0482275 (58 \#2349)
- X. Carvajal and W. Neves, Persistence of solutions to higher order nonlinear Schrödinger equation, J. Differential Equations 249 (2010), no. 9, 2214–2236. MR 2718656 (2011g:35346), DOI https://doi.org/10.1016/j.jde.2010.05.013
- Nakao Hayashi, Kuniaki Nakamitsu, and Masayoshi Tsutsumi, Nonlinear Schrödinger equations in weighted Sobolev spaces, Funkcial. Ekvac. 31 (1988), no. 3, 363–381. MR 987792 (90d:35265)
- Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR 759907 (86f:35160)
- Tosio Kato and Kyūya Masuda, Nonlinear evolution equations and analyticity. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 6, 455–467 (English, with French summary). MR 870865 (88h:34041)
- Joules Nahas, A decay property of solutions to the k-generalized KdV equation, Adv. Differential Equations 17 (2012), no. 9-10, 833–858. MR 2985676
- J. Nahas, A decay property of solutions to the mKdV equation, PhD thesis, University of California-Santa Barbara, June 2010.
- J. Nahas and G. Ponce, On the persistence properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces, Harmonic analysis and nonlinear partial differential equations, RIMS Kôkyûroku Bessatsu, B26, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 23–36. MR 2883844 (2012k:35516)
- J. Nahas and G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation, Comm. Partial Differential Equations 34 (2009), no. 10-12, 1208–1227. MR 2581970 (2011a:35498), DOI https://doi.org/10.1080/03605300903129044
- T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), no. 2, 201–222. MR 1741843 (2000m:35167)
- K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, Coupled higher-order nonlinear Schrödinger equations in nonlinear optics: Painlevé analysis and integrability, Phys. Rev. E (3) 50 (1994), no. 2, 1543–1547. MR 1381874 (96k:35171), DOI https://doi.org/10.1103/PhysRevE.50.1543
- Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Self-focusing and wave collapse. Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. MR 1696311 (2000f:35139)
- Hideo Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations 4 (1999), no. 4, 561–580. MR 1693278 (2000e:35221)
- Yoshio Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115–125. MR 915266 (89c:35143)
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Additional Information
X. Carvajal
Affiliation:
Instituto de Matemática - Universidade Federal do Rio de Janeiro UFRJ. Av. Athos da Silveira Ramos 149, Centro de Tecnologia Cidade Universitária, Ilha do Fundão, 21941-972 Rio de Janeiro, RJ, Brasil
Email:
carvajal@im.ufrj.br
W. Neves
Affiliation:
Instituto de Matemática - Universidade Federal do Rio de Janeiro UFRJ. Av. Athos da Silveira Ramos 149, Centro de Tecnologia Cidade Universitária, Ilha do Fundão, 21941-972 Rio de Janeiro, RJ, Brasil
Email:
wladimir@im.ufrj.br
Keywords:
Nonlinear dispersive equations,
nonlinear Schrödinger equation,
generalized Korteweg-de Vries equation,
initial value problem,
weighted Sobolev spaces
Received by editor(s):
September 26, 2013
Published electronically:
June 12, 2015
Article copyright:
© Copyright 2015
Brown University
The copyright for this article reverts to public domain 28 years after publication.
