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# Blow-up criteria for the inhomogeneous nonlinear Schrödinger equation

- Han Yang
^{1}and - Shihui Zhu
^{2}Email author

**2014**:55

https://doi.org/10.1186/1029-242X-2014-55

© Yang and Zhu; licensee Springer. 2014

**Received:**3 November 2013**Accepted:**20 January 2014**Published:**4 February 2014

## 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.

## Keywords

- inhomogeneous nonlinear Schrödinger equation
- blow-up criteria
- virial identity

## 1 Introduction

Recently, this type of inhomogeneous nonlinear Schrödinger equations has been widely investigated. When ${k}_{1}\le K(x)\le {k}_{2}$ with ${k}_{1},{k}_{2}>0$ and $p=2+\frac{4}{N}$, Merle [3] proved the existence and nonexistence of blow-up solutions to the Cauchy problem (1.3). When $K(x)=K(\epsilon |x|)\in {C}^{4}({\mathbb{R}}^{N})\cap {L}^{\mathrm{\infty}}({\mathbb{R}}^{N})$ with small *ε* and $p=2+\frac{4}{N}$, Fibich, Liu and Wang [2, 4] obtained the stability and instability of standing waves to the Cauchy problem (1.3).

Ginibre and Velo [5] showed the local well-posedness in ${H}^{1}({\mathbb{R}}^{N})$. Glassey [6] showed the existence of blow-up solutions when the energy is negative and $|x|{v}_{0}\in {L}^{2}({\mathbb{R}}^{N})$. Ogawa and Tsutsumi [7] obtained the existence of blow-up solutions in radial case without the restriction $|x|{v}_{0}\in {L}^{2}({\mathbb{R}}^{N})$. 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 $|x|{v}_{0}\in {L}^{2}({\mathbb{R}}^{N})$ for $p=2+\frac{4}{N}$. Lushnikov [11] and Holmer *et al.* [12] obtained some sufficient conditions for existence of blow-up for $p=4$ and $N=3$ basing on an estimate of the kinetic energy.

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 $J(t):={\int}_{{\mathbb{R}}^{N}}|x{|}^{2}|u(t,x){|}^{2}\phantom{\rule{0.2em}{0ex}}dx$, this can be interpreted as the average width of the initial distribution $|u|$. It follows from Chen and Guo’s results in [13] that we have the following proposition.

**Proposition 1.1**

*Assume that*${u}_{0}\in {H}_{r}^{1}$, $|x|{u}_{0}\in {L}^{2}$

*and the corresponding solution*$u(t,x)$

*of the Cauchy problem*(1.1)-(1.2)

*on the interval*$[0,T)$.

*Then*,

*for all*$t\in [0,T)$,

*one has*$J(t):=\int |x{|}^{2}|u(t,x){|}^{2}\phantom{\rule{0.2em}{0ex}}dx<+\mathrm{\infty}$,

*and*

one has the following theorem (see also Chen and Guo [13]).

**Theorem 1.2**

*Let*$N\ge 2$, $0<b<N-2$

*and*$\frac{2N+2b+4}{N}<p<\tilde{p}$ (

*where*$\tilde{p}=+\mathrm{\infty}$

*for*$N=2$, $\tilde{p}=\frac{2N}{N-2}+\frac{2b}{N-1}$

*for*$N\ge 3$).

*Assume*${u}_{0}\in {H}^{1}({\mathbb{R}}^{N})$

*and*$|x|{u}_{0}\in {L}^{2}({\mathbb{R}}^{N})$

*is radially symmetric*.

*If the initial data satisfies either*

- (i)$E({u}_{0})<0,$(1.8)
- (ii)$E({u}_{0})=0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{J}^{\prime}(0)<0,$(1.9)
- (iii)$E({u}_{0})>0$
*and*${J}^{\prime}(0)<-2\sqrt{2(Np-2N-2b)E({u}_{0})}J(0),$(1.10)

*then there exists* $0<T<+\mathrm{\infty}$ *such that the corresponding solution* $u(t,x)$ *blows up in finite time* *T*.

We remark that in the case $E({u}_{0})>0$, $Np-2N-2b-4\ge 0$ 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 $E({u}_{0})<0$ or ${J}^{\prime}(0)<0$. A natural question arises whether there is a sufficient condition for existence of blow-up solutions with $E({u}_{0})>0$ and ${J}^{\prime}(0)>0$.

Then we have the following theorem.

**Theorem 1.3**

*Let*$N\ge 2$, $0<b<N-2$

*and*$\frac{2N+2b+4}{N}<p<min\{\frac{2b+2N+12}{N},\tilde{p}\}$.

*Assume that*${u}_{0}\in {H}^{1}({\mathbb{R}}^{N})$

*and*$|x|{u}_{0}\in {L}^{2}({\mathbb{R}}^{N})$

*is radially symmetric*.

*If*

*where* $g(x)$ *is defined by* (1.11), *then there exists* $0<T<+\mathrm{\infty}$ *such that the corresponding solution* $u(t,x)$ *blows up in finite time* *T*.

## 2 Notations and preliminaries

In this paper, we denote ${L}^{q}({\mathbb{R}}^{N})$, ${\parallel \cdot \parallel}_{{L}^{q}({\mathbb{R}}^{N})}$, ${H}^{s}({\mathbb{R}}^{N})$ and ${\int}_{{\mathbb{R}}^{N}}\cdot \phantom{\rule{0.2em}{0ex}}dx$ by ${L}^{q}$, ${\parallel \cdot \parallel}_{{L}^{q}}$, ${H}^{s}$ and $\int \cdot \phantom{\rule{0.2em}{0ex}}dx$, respectively. ℜ*z* and ℑ*z* are the real part and imaginary part of the complex number *z*, respectively. $\overline{z}$ is denoted the complex conjugate of the complex number *z*. The various positive constants will be simply denoted by *C*.

The functional $E(u)$ 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 ${H}_{r}^{1}$, as follows.

**Proposition 2.1**

*Let*$N\ge 2$, $b\ge 0$

*and*$2+2b/(N-1)<p<\tilde{p}$ (

*where*$\tilde{p}=+\mathrm{\infty}$

*for*$N=2$, $\tilde{p}=\frac{2N}{N-2}+\frac{2b}{N-1}$

*for*$N\ge 3$).

*For any*${u}_{0}\in {H}_{r}^{1}$,

*there exists a unique solution*$u(t,x)$

*of the Cauchy problem*(1.1)-(1.2)

*on the maximal time interval*$[0,T)$

*such that*$u(t,x)\in C([0,T);{H}_{r}^{1})$

*and either*$T=+\mathrm{\infty}$ (

*global existence*),

*or*$T<+\mathrm{\infty}$

*and*${lim}_{t\to T}{\parallel u(t,x)\parallel}_{{H}_{r}^{1}}=+\mathrm{\infty}$ (

*blow*-

*up*).

*Furthermore*,

*for all*$t\in [0,T)$, $u(t,x)$

*satisfies the following conservation laws*:

- (i)
*Conservation of mass*: ${\parallel u(t)\parallel}_{2}={\parallel {u}_{0}\parallel}_{2}$. - (ii)
*Conservation of energy*: $E(u(t,x))=E({u}_{0})$.

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*$V(t)>0$

*is the positive solution of the following differential equation*:

*and there exists*$0<{T}_{1}<+\mathrm{\infty}$

*such that*${lim}_{t\to {T}_{1}}V(t)=0$,

*then for the solution*$\tilde{V}>0$

*of the following differential equation*:

*there exists* $0<{T}_{2}\le {T}_{1}<+\mathrm{\infty}$ *such that* ${lim}_{t\to {T}_{2}}\tilde{V}(t)=0$.

*Proof* Since the function $-{h}^{2}(t)$ is non-positive, which pulls $\tilde{V}(t)$ to zero more quickly than $V(t)$ (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

*t*, we get the corresponding mechanical energy

*p*,

*b*and

*N*we see that $U(A)$ achieves its maximum ${U}_{max}$ at ${A}_{max}$ with

- (a)
$\tilde{\epsilon}(0)<1$ and $\tilde{A}(0)<1$,

- (b)
$\tilde{\epsilon}(0)\ge 1$ and ${\tilde{A}}_{s}(0)<0$.

## Declarations

### Acknowledgements

This work is supported partly by the National Natural Science Foundation of P.R. China grants 11226162 and 11371267.

## Authors’ Affiliations

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