DIVERGENT SOLUTION TO THE NONLINEAR SCHR&#214;DINGER EQUATION WITH THE COMBINED POWER-TYPE NONLINEARITIES

Jing Li; Boling Guo

doi:10.11948/2017017

2017 Volume 7 Issue 1

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Jing Li, Boling Guo. DIVERGENT SOLUTION TO THE NONLINEAR SCHRÖDINGER EQUATION WITH THE COMBINED POWER-TYPE NONLINEARITIES[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 249-263. doi: 10.11948/2017017

Citation:

Jing Li, Boling Guo. DIVERGENT SOLUTION TO THE NONLINEAR SCHRÖDINGER EQUATION WITH THE COMBINED POWER-TYPE NONLINEARITIES[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 249-263. doi: 10.11948/2017017

DIVERGENT SOLUTION TO THE NONLINEAR SCHRÖDINGER EQUATION WITH THE COMBINED POWER-TYPE NONLINEARITIES

Jing Li¹,
Boling Guo²

1 School of Mathematics and Computing Science, Changsha University of Science and Technology, 410114 Changsha, China;
2 Institute of Applied physics and computational Mathematics, 10088 Beijing, China

Fund Project:

Abstract

Abstract

In this paper, we consider the Cauchy problem for the nonlinear Schrödinger equation with combined power-type nonlinearities, which is masscritical/supercr-itical, and energy-subcritical. Combing Du, Wu and Zhang' argument with the variational method, we prove that if the energy of the initial data is negative (or under some more general condition), then the H¹-norm of the solution to the Cauchy problem will go to infinity in some finite time or infinite time.
- Nonlinear Schrödinger equation /
- combined power-type nonlinearities /
- blow-up
MSC: 35Q41;35B44

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References

[1]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations I-Ⅱ, Archive for Rational Mechanics and Analysis, 82(1983)(4), 313-345. Google Scholar
[2]	H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non lineáires, Comptes Rendus de l Acadmie des Sciences-Series I-Mathematics, 293(1981)(9), 489-492. Google Scholar
[3]	D. Cao and Q. Guo, Divergent solutions to the 5D Hartree equations, Colloquium Mathematicum, 125(2011)(2), 255-287. Google Scholar
[4]	T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H^s, Nonlinear Analysis:Theory Methods & Applications, 14(1990)(10), 807-836. Google Scholar
[5]	T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 4(2003)(3), 635-637. Google Scholar
[6]	D. Du, Y. Wu and K. Zhang, On Blow-up criterion for the Nonlinear Schrödinger Equation, arXiv:1309.6782, 2013. Google Scholar
[7]	R. Fukuizui, Remarks on the stable standing waves for nonlinear Schrödinger equations with double power nonlinearity, Advances in Mathematical Sciences and Applications, 13(2003)(2), 549-564. Google Scholar
[8]	Z. H. Gan, B. L. Guo and J. Zhang, Sharp threshold of global existence for the generalized Davey-Stewartson system in R2, Communications on Pure & Applied Analysis, 8(2009)(3), 913-922. Google Scholar
[9]	L. Glangetas and F. Merle, A geometrical approach of existence of blow up solutions in H1 for nonlinear Schrödinger equation, Prepublication Univ. P.M. Curie, R95031, 1995. Google Scholar
[10]	R. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equation, Journal of Mathematical Physics, 18(1977)(9), 1794-1797. Google Scholar
[11]	B. L. Guo and B. X. Wang, The Cauchy problem for Davey-Stewartson systems, Communications on Pure & Applied Mathematics, 52(1999)(12), 1477-1490. Google Scholar
[12]	Q. Guo, Nonscattering solutions to the L²-supercritical NLS equations, arXiv:1101.2271, 2011. Google Scholar
[13]	J. Holmer and S. Roudenko, Divergence of infinite-variance nonradial solutions to the 3d NLS equation, Communications in Partial Differential Equations, 35(2009)(5), 878-905. Google Scholar
[14]	C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case, Inventiones Mathematicae, 166(2006)(3), 645-675. Google Scholar
[15]	Y. S. Li and B. L. Guo, Existence and decay of weak solutions to degenerate Davey-Stewartson equations, Acta Mathematica Scientia (English Edition), 22(2002)(3), 302-310. Google Scholar
[16]	C. Miao, G. Xu and L. Zhao, The dynamics of the 3D radial NLS with the combined terms, Communications in Mathematical Physics, 318(2013)(3), 767-808. Google Scholar
[17]	T. Ogawa and Y. Tsutsumi, Blow-up of H¹ solution for the nonlinear Schrödinger equation, Journal of Differential Equations, 92(1991)(2), 317-330. Google Scholar
[18]	M. Ohta, Instability of standing waves for the generalized Daey-Stewartson system. Annales de l'IHP Physique théorique, 62(1995)(1), 69-80. Google Scholar
[19]	M. Ohta, Stability and instability of standing waves for one-dimensional nonlinear Schrödinger equations with double power nonlinearity, Kodai Mathematical Journal, 18(1995)(1), 68-74. Google Scholar
[20]	M. Ohta and T. Yamaguchi, Strong instability of standing waves for nonlinear schrodinger equations with double power nonlinearity, arXiv:1411.2620, 2014. Google Scholar
[21]	W. Strauss, Existence of solitary waves in higher dimensions, Communications in Mathematical Physics, 55(1977)(2), 149-162. Google Scholar
[22]	T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Communications in Partial Differential Equations, 32(2007)(8), 1281-1343. Google Scholar
[23]	B. X. Wang and B. L. Guo, On the initial value problem and scattering of solutions for the generalized Davey-Stewartson systems, Science China Ser. A, 44(2001)(8), 994-1002. Google Scholar