Abstract
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.
MSC:35Q55, 35B44.
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1 Introduction
In this paper, we study the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation (INSE)
where ; is the Laplace operator in ; : is the complex valued function and ; the parameter and (we use the convention: for , for ); is the space dimension. A few years ago, it was suggested that stable high power propagation can be achieved in a plasma by sending a preliminary laser beam that creates a channel with a reduced electron density, and thus reduces the nonlinearity inside the channel (see [1, 2]). In this case, the beam propagation can be modeled by the inhomogeneous nonlinear Schrödinger equation in the following form:
Recently, this type of inhomogeneous nonlinear Schrödinger equations has been widely investigated. When with and , Merle [3] 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).
We recall some known results on the blow-up solutions for the classical nonlinear Schrödinger equation
Ginibre and Velo [5] showed the local well-posedness in . Glassey [6] showed the existence of blow-up solutions when the energy is negative and . Ogawa and Tsutsumi [7] obtained the existence of blow-up solutions in radial case without the restriction . Weinstein [8] and Zhang [9] obtained the sharp conditions of global existence for critical and supercritical nonlinearity. Merle and Raphaël [10] showed the existence of blow-up solutions without for . Lushnikov [11] and Holmer et al. [12] obtained some sufficient conditions for existence of blow-up for and basing on an estimate of the kinetic energy.
In this paper, we study blow-up criteria for the Cauchy problem (1.1)-(1.2) with , where the nonlinearity includes an unbounded potential . We note that (1.1) has scaling: is a solution if is a solution. The scale-invariant Lebesgue norm for this equation is -norm, where . Since , we have and we may call (1.1) a class of Schrödinger equations with -super critical nonlinearity. Chen and Guo [13] and Chen [14] showed the local well-posedness of the Cauchy problem (1.1)-(1.2) in , where is the set of radially symmetric functions in . Moreover, satisfies the following conservation laws:
and
Chen and Guo [13] 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 [13] that we have the following proposition.
Proposition 1.1 Assume that , and the corresponding solution of the Cauchy problem (1.1)-(1.2) on the interval . Then, for all , one has ,
and
In the case , it follows from the last relation that
and for the positive-definite quantity becomes negative over a finite time by virtue of the above inequality. This means that a singularity appears in the solution of the given INSE. Indeed, applying Weinstein’s arguments [8] and the classical analysis identity
one has the following theorem (see also Chen and Guo [13]).
Theorem 1.2 Let , and (where for , for ). Assume and is radially symmetric. If the initial data satisfies either
-
(i)
(1.8)
-
(ii)
(1.9)
-
(iii)
and
(1.10)
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 [13] if either or . A natural question arises whether there is a sufficient condition for existence of blow-up solutions with and .
In the present paper, motivated by the studies of the classical nonlinear Schrödinger equation (see [8, 11, 12]), we use variational characteristic of second-order derivatives of the virial identity to catch up with the information of , and we obtain a new sufficient condition for the existence of blow-up solutions to the inhomogeneous nonlinear Schrödinger equation (1.1). More precisely, let
Then we have the following theorem.
Theorem 1.3 Let , and . Assume that and is radially symmetric. If
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.
For the Cauchy problem (1.1)-(1.2), the space we work in is
which is a Hilbert space. Moreover, we define the energy functional in by
The functional is well-defined according to the Sobolev embedding theorem (see [15]). Chen and Guo [13] and Chen [14] showed the local well-posedness for the Cauchy problem (1.1)-(1.2) in , as follows.
Proposition 2.1 Let , and (where for , for ). For any , there exists a unique solution of the Cauchy problem (1.1)-(1.2) on the maximal time interval such that and either (global existence), or and (blow-up). Furthermore, for all , satisfies the following conservation laws:
-
(i)
Conservation of mass: .
-
(ii)
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 [11].
Lemma 2.2 If is the positive solution of the following differential equation:
and there exists such that , then for the solution of the following differential equation:
there exists such that .
Proof Since the function is non-positive, which pulls to zero more quickly than (see also [11]), one sees that the conclusion in Lemma 2.2 is true by the classical analysis identity (1.7). □
3 Proof of Theorem 1.3
Since , we have by the local well-posedness. Taking , by Proposition 1.1, we get and
It follows from some calculations that
Then we get , and by the fact that we deduce that
Injecting (3.2) into (3.1), by the conservation laws, we deduce that
Letting and rewriting (3.3) to remove the last term with , we have
which has a simple mechanism analogy. Multiplying in (3.4) and integrating with the time variable t, we get the corresponding mechanical energy
where
Restricting ourselves to the case , and according to the assumptions on p, b and N we see that achieves its maximum at with
and
To facilitate the rest of the analysis, we introduce a rescaling. Define and by the relations
Thus, by (3.4), satisfies the following differential inequality:
Applying Lemma 2.2, if we show that for the positive solution of the following differential equation:
there exists a time and , then for the positive solution of the differential inequality (3.6), there exists a time and . Indeed, setting
we see that (3.5) converts to
It is obvious that the maximum of is 1, which is attained by the maximum at . By the variational characteristic of , we claim that under one of the following conditions, vanishes in finite time, so does (which implies the solution blows up in finite time):
-
(a)
and ,
-
(b)
and .
Indeed, it follows from (3.7) and (3.9) that . (a) If , then and for (maximal existence interval). By the assumption , we deduce that for . Moreover, we have
Using (3.7) and the classical analysis identity
we see that vanishes in finite time. On the other hand, for the second case (b), if , then . If follows from (3.9) that , which implies that and
Using (3.7) and the classical analysis identity (3.10), we deduce that if , then vanishes in finite time. This completes the proof of claim (a) and (b). Now, we return to the proof of Theorem 1.3. If we define , then (3.9) is equal to
Taking
we see that
Thus condition (a) is true if and only if
and condition (b) is true if and only if . Collecting the above two conditions, we deduce that the solution blows up in finite time provided
Finally, substituting back , we see that
which implies that (3.12) is equivalent to
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Acknowledgements
This work is supported partly by the National Natural Science Foundation of P.R. China grants 11226162 and 11371267.
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HY participated in the design of the study, SZ studied the virial identity, participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.
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Yang, H., Zhu, S. Blow-up criteria for the inhomogeneous nonlinear Schrödinger equation. J Inequal Appl 2014, 55 (2014). https://doi.org/10.1186/1029-242X-2014-55
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DOI: https://doi.org/10.1186/1029-242X-2014-55