Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T14:57:13.687Z Has data issue: false hasContentIssue false

GLOBAL WELL-POSEDNESS AND INSTABILITY OF A NONLINEAR SCHRÖDINGER EQUATION WITH HARMONIC POTENTIAL

Published online by Cambridge University Press:  14 October 2014

T. SAANOUNI*
Affiliation:
University of Tunis El Manar, Faculty of Sciences of Tunis, LR03ES04 Partial Differential Equations and Applications, 2092 Tunis, Tunisia email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is concerned with the Cauchy problem for a nonlinear Schrödinger equation with a harmonic potential and exponential growth nonlinearity in two space dimensions. In the defocusing case, global well-posedness is obtained. In the focusing case, existence of nonglobal solutions is discussed via potential-well arguments.

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Adachi, S. and Tanaka, K., ‘Trudinger type inequalities in ℝN and their best exponent’, Proc. Amer. Math. Soc. 128(7) (1999), 20512057.CrossRefGoogle Scholar
Adams, R. A., Sobolev Spaces (Academic Press, New York, 1975).Google Scholar
Bradley, C. C., Sackett, C. A. and Hulet, R. G., ‘Bose–Einstein condensation of lithium: observation of limited condensate number’, Phys. Rev. Lett. 78 (1997), 985989.CrossRefGoogle Scholar
Carles, R., ‘Remarks on the nonlinear Schrödinger equation with harmonic potential’, Ann. Henri Poincaré 3 (2002), 757772.CrossRefGoogle Scholar
Carles, R., ‘Critical nonlinear Schrödinger equations with and without harmonic potential’, Math. Models Methods Appl. Sci. 12 (2002), 15131523.CrossRefGoogle Scholar
Carles, R., ‘Nonlinear Schrödinger equation with time dependent potential’, Commun. Math. Sci. 9(4) (2011), 937964.CrossRefGoogle Scholar
Cazenave, T., An Introduction to Nonlinear Schrödinger Equations, Textos de Metodos Matematicos, 26 (Instituto de Matematica – UFRJ, Brazil, 1996).Google Scholar
Colliander, J., Ibrahim, S., Majdoub, M. and Masmoudi, N., ‘Energy critical NLS in two space dimensions’, J. Hyperbolic Differ. Equ. 6 (2009), 549575.CrossRefGoogle Scholar
Dalfovo, F., Giorgini, S., Pitaevskii, P. L. and Stringari, S., ‘Theory of Bose–Einstein condensation in trapped gases’, Rev. Modern Phys. 71(3) (1999), 463512.CrossRefGoogle Scholar
Fujiwara, D., ‘A construction of the fundamental solution for the Schrödinger equation’, J. Anal. Math. 35 (1979), 4196.CrossRefGoogle Scholar
Fujiwara, D., ‘Remarks on the convergence of the Feynman path integrals’, Duke Math. J. 47(3) (1980), 559600.CrossRefGoogle Scholar
Ibrahim, S., Majdoub, M. and Masmoudi, N., ‘Double logarithmic inequality with a sharp constant’, Proc. Amer. Math. Soc. 135(1) (2007), 8797.CrossRefGoogle Scholar
Kenig, C. and Merle, F., ‘Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case’, Invent. Math. 166 (2006), 645675.CrossRefGoogle Scholar
Lam, J. F., Lippman, B. and Trappert, F., ‘Self trapped laser beams in plasma’, Phys. Fluids 20 (1997), 11761179.CrossRefGoogle Scholar
Mahouachi, O. and Saanouni, T., ‘Global well-posedness and linearization of a semilinear wave equation with exponential growth’, Georgian Math. J. 17 (2010), 543562.CrossRefGoogle Scholar
Mahouachi, O. and Saanouni, T., ‘Well and ill posedness issues for a 2D wave equation with exponential nonlinearity’, J. Partial Differ. Equ. 24(4) (2011), 361384.Google Scholar
Moser, J., ‘A sharp form of an inequality of N. Trudinger’, Indiana Univ. Math. J. 20 (1971), 10771092.CrossRefGoogle Scholar
Nakamura, M. and Ozawa, T., ‘Nonlinear Schrödinger equations in the Sobolev space of critical order’, J. Funct. Anal. 155 (1998), 364380.CrossRefGoogle Scholar
Oh, Y. G., ‘Cauchy problem and Ehrenfest’s law of nonlinear Schrödinger equations with potentials’, J. Differential Equations 81 (1989), 255274.CrossRefGoogle Scholar
Payne, L. E. and Sattinger, D. H., ‘Saddle points and instability of nonlinear hyperbolic equations’, Israel J. Math. 22 (1975), 273303.CrossRefGoogle Scholar
Pitaevskii, L. P., ‘Dynamics of collapse of a confined Bose gas’, Phys. Lett. A 221 (1996), 1418.CrossRefGoogle Scholar
Ruf, B., ‘A sharp Moser–Trudinger type inequality for unbounded domains in ℝ2’, J. Funct. Anal. 219 (2004), 340367.CrossRefGoogle Scholar
Saanouni, T., ‘Global well-posedness and scattering of a 2D Schrödinger equation with exponential growth’, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 441462.CrossRefGoogle Scholar
Saanouni, T., ‘Scattering of a 2D Schrödinger equation with exponential growth in the conformal space’, Math. Methods Appl. Sci. 33 (2010), 10461058.CrossRefGoogle Scholar
Saanouni, T., ‘Decay of solutions to a 2D Schrödinger equation with exponential growth’, J. Partial Differ. Equ. 24(1) (2011), 3754.Google Scholar
Saanouni, T., ‘Blowing-up semilinear wave equation with exponential nonlinearity in two space dimensions’, Proc. Indian Acad. Sci. Math. Sci. 123(3) (2013), 365372.CrossRefGoogle Scholar
Saanouni, T., ‘Remarks on the semilinear Schrödinger equation’, J. Math. Anal. Appl. 400 (2013), 331344.CrossRefGoogle Scholar
Saanouni, T., ‘Global well-posedness and instability of a 2D Schrödinger equation with harmonic potential in the conformal space’, J. Abstr. Differ. Equ. Appl. 4(1) (2013), 2342.Google Scholar
Saanouni, T., ‘Global well-posedness of a damped Schrödinger equation in two space dimensions’, Math. Methods Appl. Sci. 37(4) (2014), 488495, (published online).CrossRefGoogle Scholar
Saanouni, T., ‘A note on the instability of a focusing nonlinear damped wave equation’, Math. Methods Appl. Sci. (2014), published online (wileyonlinelibrary.com), doi:10.1002/mma.3044.CrossRefGoogle Scholar
Tao, T., Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Series in Mathematics, 106 (American Mathematical Society, Providence, RI, 2006).CrossRefGoogle Scholar
Trudinger, N. S., ‘On imbedding into Orlicz spaces and some applications’, J. Math. Mech. 17 (1967), 473484.Google Scholar
Tsurumi, T. and Wadati, M., ‘Collapses of wave functions in multidimensional nonlinear Schrödinger equations under harmonic potential’, Phys. Soc. Japan 66 (1997), 30313034.CrossRefGoogle Scholar
Weinstein, M. I., ‘Nonlinear Schrödinger equations and sharp interpolation estimates’, Comm. Math. Phys. 87 (1983), 567576.CrossRefGoogle Scholar
Zakharov, V. E., ‘Collapse of Langmuir waves’, Sov. Phys. JETP 23 (1996), 10251033.Google Scholar
Zhang, J., ‘Stability of attractive Bose–Einstein condensates’, J. Stat. Phys. 101 (2000), 731746.CrossRefGoogle Scholar