Blow-up criteria for the inhomogeneous nonlinear Schrödinger equation
© Yang and Zhu; licensee Springer. 2014
Received: 3 November 2013
Accepted: 20 January 2014
Published: 4 February 2014
In this paper, using the variational characteristic of the virial identity and a new estimate of the kinetic energy, we obtain a new sufficient condition for the existence of blow-up solutions.
Recently, this type of inhomogeneous nonlinear Schrödinger equations has been widely investigated. When with and , Merle  proved the existence and nonexistence of blow-up solutions to the Cauchy problem (1.3). When with small ε and , Fibich, Liu and Wang [2, 4] obtained the stability and instability of standing waves to the Cauchy problem (1.3).
Ginibre and Velo  showed the local well-posedness in . Glassey  showed the existence of blow-up solutions when the energy is negative and . Ogawa and Tsutsumi  obtained the existence of blow-up solutions in radial case without the restriction . Weinstein  and Zhang  obtained the sharp conditions of global existence for critical and supercritical nonlinearity. Merle and Raphaël  showed the existence of blow-up solutions without for . Lushnikov  and Holmer et al.  obtained some sufficient conditions for existence of blow-up for and basing on an estimate of the kinetic energy.
Chen and Guo  also showed the sharp conditions of blow-up and global existence of solutions to the Cauchy problem (1.1)-(1.2) by the cross-constrained variational arguments. On the other hand, letting , this can be interpreted as the average width of the initial distribution . It follows from Chen and Guo’s results in  that we have the following proposition.
one has the following theorem (see also Chen and Guo ).
then there exists such that the corresponding solution blows up in finite time T.
We remark that in the case , both collapse and spreading of the initial disturbance are possible. Although the INSE is no longer applicable near the formation point of a singularity and dissipative or some other limiting mechanism come to play. It is very important to be able to predict the presence or absence of collapse for different classes of initial conditions. The sufficient conditions for existence of blow-up solutions are given in  if either or . A natural question arises whether there is a sufficient condition for existence of blow-up solutions with and .
Then we have the following theorem.
where is defined by (1.11), then there exists such that the corresponding solution blows up in finite time T.
2 Notations and preliminaries
In this paper, we denote , , and by , , and , respectively. ℜz and ℑz are the real part and imaginary part of the complex number z, respectively. is denoted the complex conjugate of the complex number z. The various positive constants will be simply denoted by C.
The functional is well-defined according to the Sobolev embedding theorem (see ). Chen and Guo  and Chen  showed the local well-posedness for the Cauchy problem (1.1)-(1.2) in , as follows.
Conservation of mass: .
Conservation of energy: .
In addition, by some basic calculations, we have the following lemma, which gives further insight in the dynamic criterion for collapse proposed by Lushnikov in .
there exists such that .
Proof Since the function is non-positive, which pulls to zero more quickly than (see also ), one sees that the conclusion in Lemma 2.2 is true by the classical analysis identity (1.7). □
3 Proof of Theorem 1.3
This work is supported partly by the National Natural Science Foundation of P.R. China grants 11226162 and 11371267.
- Gill TS: Optical guiding of laser beam in nonuniform plasma. Pramana J. Phys. 2000, 55: 842–845.Google Scholar
- Liu CS, Tripathi VK: Laser guiding in an axially nonuniform plasma channel. Phys. Plasmas 1994, 1: 3100–3103. 10.1063/1.870501View ArticleGoogle Scholar
- Merle F:Nonexistence of minimal blow up solutions of equations in . Ann. IHP, Phys. Théor. 1996, 64: 33–85.MathSciNetMATHGoogle Scholar
- Liu Y, Wang XP, Wang K: Instability of standing waves of the Schrödinger equations with inhomogeneous nonlinearities. Trans. Am. Math. Soc. 2006, 358: 2105–2122. 10.1090/S0002-9947-05-03763-3View ArticleMATHGoogle Scholar
- Ginibre J, Velo G: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 1979, 32: 1–32. 10.1016/0022-1236(79)90076-4MathSciNetView ArticleMATHGoogle Scholar
- Glassey RT: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys. 1977, 18: 1794–1797. 10.1063/1.523491MathSciNetView ArticleMATHGoogle Scholar
- Ogawa T, Tsutsumi Y:Blow-up of solution for the nonlinear Schrödinger equation. J. Differ. Equ. 1991, 92: 317–330. 10.1016/0022-0396(91)90052-BMathSciNetView ArticleMATHGoogle Scholar
- Weinstein MI: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 1983, 87: 567–576. 10.1007/BF01208265View ArticleMATHGoogle Scholar
- Zhang J: Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations. Nonlinear Anal. 2002, 48: 191–207. 10.1016/S0362-546X(00)00180-2MathSciNetView ArticleMATHGoogle Scholar
- Merle F, Raphaël P: Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. Math. 2005, 16: 157–222.View ArticleMATHGoogle Scholar
- Lushnikov PM: Dynamic criterion for collapse. JETP Lett. 1995, 62: 461–467.Google Scholar
- Holmer J, Platte R, Roudenko S: Blow-up criteria for the 3D cubic nonlinear Schrödinger equation. Nonlinearity 2010, 23: 977–1030. 10.1088/0951-7715/23/4/011MathSciNetView ArticleMATHGoogle Scholar
- Chen JQ, Guo BL: Sharp global existence and blowing up results for inhomogeneous Schrödinger equations. Discrete Contin. Dyn. Syst., Ser. B 2007, 8: 357–367.MathSciNetView ArticleMATHGoogle Scholar
- Chen JQ: On a class of nonlinear inhomogeneous Schrödinger equation. J. Appl. Math. Comput. 2010, 32: 237–253. 10.1007/s12190-009-0246-5MathSciNetView ArticleMATHGoogle Scholar
- Cazenave T Courant Lecture Notes in Mathematics 10. In Semilinear Schrödinger Equations. Am. Math. Soc., Providence; 2003.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.