A new blow-up criterion for the nonhomogeneous nonlinear Schrödinger equation

LI Shuangshuang

doi:10.3969/j.issn.1000-5641.201911029

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Jul. 2020

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LI Shuangshuang. A new blow-up criterion for the nonhomogeneous nonlinear Schrödinger equation[J]. Journal of East China Normal University (Natural Sciences), 2020, (4): 64-71. doi: 10.3969/j.issn.1000-5641.201911029

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LI Shuangshuang. A new blow-up criterion for the nonhomogeneous nonlinear Schrödinger equation[J]. Journal of East China Normal University (Natural Sciences), 2020, (4): 64-71. doi: 10.3969/j.issn.1000-5641.201911029

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A new blow-up criterion for the nonhomogeneous nonlinear Schrödinger equation

doi: 10.3969/j.issn.1000-5641.201911029

LI Shuangshuang

College of Mathematics and Statistics, Northwest Normal University, Lanzhou　730070, China

Received Date: 2019-06-26
Available Online: 2020-07-20
Publish Date: 2020-07-25

Abstract

Abstract

In this paper, the existence of blow-up solutions for the nonhomogeneous nonlinear Schrödinger equation is studied. First, a class of invariant sets is constructed and then the optimal Gagliardo-Nirenberg type inequality is applied; careful analysis is used to prove that for any large $\mu$, there exists $u_{0}\in H^{1}$ so that $E(u_{0})=\mu$ and the solution $u(t,x)$ with $u_{0}$ as an initial value blows up in finite time. This result supplements the existing content in the literature [1].
- nonhomogeneous nonlinear Schrödinger equation,
- invariant set,
- blow-up

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References(15)

References

[1]	FARAH L G. Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equations [J]. J Evol Equ, 2016, 16(1): 193-208. DOI: 10.1007/s00028-015-0298-y.
[2]	LIU C S, TRIPATHI V K. Laser guiding in an axially nonumiform plasma channel [J]. Physics of Plasmas, 1994, 1(9): 3100-3103. DOI: 10.1063/1.870501.
[3]	CHEN J Q, GUO B L. Sharp global existence and blowing up results for inhomogeneous Schrödinger equations [J]. Discrete Cont Dyn-B, 2007, 8(2): 357-367. DOI: 10.3934/dcdsb.2007.8.357.
[4]	HMIDI T, KERAANI S. On the blowup theory for the critical nonlinear Schrödinger equations [J]. Int Math Res Notices, 2005, 21(21): 2815-2828. DOI: 10.1155/IMRN.2005.2815.
[5]	GILL T S. Optical guiding of laser beam in nonuniform plasma [J]. Pramana-J Phys, 2000, 55(5/6): 835-842. DOI: 10.1007/s12043-000-0051-z.
[6]	GENOUD F, STUART C A. Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves [J]. Discrete Cont Dyn-A, 2008, 21(1): 137-186. DOI: 10.3934/dcds.2008.21.137.
[7]	ZHANG J. Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations [J]. Nonlinear Anal TMA, 2002, 48(2): 191-207. DOI: 10.1016/S0362-546X(00)00180-2.
[8]	GLASSEY R T. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations [J]. J Math Phys, 1977, 18(9): 1794-1797. DOI: 10.1063/1.523491.
[9]	MERlE F. Nonexistence of minimal blow-up solutions of equation \scriptsize $ {\rm{i}}u_{t}=-\Delta u-k(x)\|u\|^{\frac{4}{N}}u $ \normalsize in \scriptsize $ \mathbb{R}^{N} $ \normalsize [J]. Annales de l’lHP Physique théorique, 1996, 64(1): 33-85.
[10]	PAPHAEL P, SZEFTEl J. Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass-critical NLS [J]. J Amer Math Soc, 2011, 24(2): 471-546. DOI: 10.1090/S0894-0347-2010-00688-1.
[11]	CHEN J Q. On a class of nonlinear inhomogeneous Schrödinger equations [J]. JAMC, 2010, 32(1): 237-253. DOI: 10.1007/s12190-009-0246-5.
[12]	YANG L Y, LI X G. Global well-posedness and blow-up for the hartree equation [J]. Acta Math Sci, 2017, 37(4): 941-948. DOI: 10.1016/S0252-9602(17)30049-8.
[13]	YUE Z T, LI X G, ZHANG J. A new blow-up criterion for Gross-Pitaevskii equation [J]. Appl Math Lett, 2016, 62: 16-22. DOI: 10.1016/j.aml.2016.06.007.
[14]	CAZENAVE T. Semilinear Schrödinger Equations [M]. New York: American Mathematical Society, 2003.
[15]	YANAGIDA E. Uniqueness of positive radial solutions of \scriptsize $ \Delta u+g(r)u+h(r)u^{p}=0 $ \normalsize in \scriptsize $ \mathbb{R}^{N} $ \normalsize [J]. Arch Ration Mech An, 1991, 115(3): 257-274. DOI: 10.1007/BF00380770.