- Open Access
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.
- inhomogeneous nonlinear Schrödinger equation
- blow-up criteria
- virial identity
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 Although the INSE is no longer applicable near the formation point of a singularity and dissipative or some other limiting mechanism come to play.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.
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). □
This work is supported partly by the National Natural Science Foundation of P.R. China grants 11226162 and 11371267.
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