- Research
- Open Access
- Published:

# Global well-posedness for nonlinear fourth-order Schrödinger equations

*Boundary Value Problems*
**volume 2016**, Article number: 25 (2016)

## Abstract

This paper studies a class of nonlinear fourth-order Schrödinger equations. By constructing a variational problem and the so-called invariant of some sets, we get global existence and nonexistence of the solutions.

## Introduction

This paper concerns the initial value problems for the nonlinear fourth-order Schrödinger equations

where \(0< p<\frac{8}{(n-4)^{+}}\) (we use the convention: \(\frac {8}{(n-4)^{+}}=+\infty\) when \(2\leq n\leq4\); \(\frac{8}{(n-4)^{+}}=\frac {8}{n-4}\) when \(n\geq5\)), \(u(t,x):\mathbb{R}\times\mathbb {R}^{n}\rightarrow\mathbb{C}\), \(n\geq2\), is the unknown function and Δ is the Laplace operator, \(t\in[0,+\infty)\).

Problem (1.1) was first introduced by Karpman [1]. Karpman and Shagalov [2] considered the conditions for existence and stability of solutions about the fourth-order Schrödinger equation

where *p* is an integer and

Pausader [3] established the global well-posedness for the energy critical fourth-order Schrödinger equation

in the radial case by Strichartz-type estimates, while a specific nonlinear fourth-order Schrödinger equation as above (1.3) had been recently discussed by Fibich *et al.* [4]. They described various properties of the equation in the subcritical regime.

Moreover, for \(f(|u|^{2})u=|u|^{p-1}u\) one could also consider the focusing equation

and proved that the solutions blow up in finite time for large data [5, 6].

Motivated by the above works, Pausader and Xia [7] proved the scattering theory for the defocusing fourth-order Schrödinger equation

in low spatial dimensions (\(1\leq n \leq4\)) by a virial-type estimate and Morawetz-type estimate.

Recently, Wang [8] proved the small data scattering and large data local well-posedness for the fourth-order nonlinear problem

where \(\mu=\pm1\), in critical \(H^{s_{c}}\) space and in particular, for some \(s_{c} \leq0\) by Fourier restriction theory and Strichartz-type estimates, but the sharp conditions of the global existence and blow up for the problem by potential well theory is still not considered for \(\mu=1\). In this paper we try to solve this problem by a concavity method and potential well theory. Recently, the concavity method and potential well theory were applied by Shen *et al.* [9] to study the initial boundary value problem for fourth-order wave equations with nonlinear strain and source terms at high energy level. For other related results, we refer the reader to [10–18].

The plan of this paper is as follows. In the second section, we state some propositions, lemmas, and definitions and prove some invariant sets. In the third section, we state the sharp condition for the global existence and nonexistence of problem (1.1).

Throughout this paper, the \(H^{2}(\mathbb{R}^{n})\)-norm will be designated by \(\|\cdot\|_{H^{2}}\), also, the \(L^{p}(\mathbb{R}^{n})\)-norm will be denoted by \(\|\cdot\|_{L^{p}}\) (if \(p=2\), \(\|\cdot\|_{L^{2}}\) is denoted \(\|\cdot\|\)). For simplicity, hereafter, \(\int_{\mathbb {R}^{n}}\cdot\,dx\) is denoted ∫⋅.

## Preliminaries

For problem (1.1), we define the energy space in the course of nature by

### Proposition 2.1

*Let*
\(u_{0}\in H\). *Then there exists a unique solution*
*u*
*of the Cauchy problem* (1.1) *in*
\(C([0, T]; H)\)
*for some*
\(T\in(0,\infty]\) (*maximal existence time*). *Furthermore*, *we can get alternatives*: \(T =\infty\)
*or*
\(T<\infty\)
*and*

*Moreover*, *u*
*satisfies*

From [19, 20], we can get the following lemma.

### Lemma 2.2

*Suppose*
\(u_{0}\in H\), \(u\in C([0,T); H)\)
*be a solution to problem* (1.1). *Let*
\(J(t)=\int|x|^{2}|u|^{2}\), *then*

Furthermore, we consider the following steady-state equation:

For any solution of (2.5), we define the following functionals:

When \(\varphi_{0}\in H\) and *φ* are a solution of problem (1.1) in \(C([0,T]; H)\), we have

and we define the set

Now, we study the following constrained variational problem:

By a similar argument to [21], we get the following lemmas.

### Lemma 2.3

*Solution of* (2.5) *belongs to*
*M*.

### Proof

Let \(\varphi(x)\) be a solution of steady-state equation (2.5). Then we get

from which

Hence \(\varphi\in M\). □

### Lemma 2.4

*When*
\(\varphi(x)\in M\), *we have*
\(d>0\).

### Proof

By (2.6) and (2.7), on *M* we get

Combined with (2.9), we obtain the conclusion. □

### Lemma 2.5

*Let*
\(\varphi\in H \), \(\lambda>0\), *and*
\(\varphi_{\lambda}(x)=\lambda \varphi(x)\). *Then there exists a unique*
\(\lambda^{*}>0\) (*depending on*
*φ*) *such that*
\(I(\varphi_{\lambda^{*}})=0\)
*and*
\(I(\varphi _{\lambda})>0\), *for*
\(\lambda\in(0, \mu)\); \(I(\varphi_{\lambda})<0\), *for*
\(\lambda>\lambda^{*}\). *Furthermore*, \(P(\varphi_{\lambda^{*}})\geq P(\varphi_{\lambda})\), *for any*
\(\lambda>0\).

### Proof

From \(\varphi_{\lambda}=\lambda\varphi\), (2.6) and (2.7), we have

and

Furthermore, there exists a unique positive constant \(\lambda^{*}>0\) (depending on *φ*) such that \(I(\varphi_{\lambda^{*}}) = 0\) and we can easily see that

and

Combining

with

we obtain

This completes the proof of the lemma. □

Now we discuss the invariant sets of solution for problem (1.1).

### Theorem 2.6

*Let*

*If*
\(u_{0}\in V\), *then the solution*
\(u(x,t)\)
*of problem* (1.1) *also belongs to*
*V*
*for any*
*t*
*in the interval*
\([0, T)\).

Notice that on the basis of Theorem 2.6 one says that *V* is an invariant set of problem (1.1).

### Proof

Let \(u_{0}\in V\). By Proposition 2.1 there exists a unique \(u(x, t)\in C([0, T); H)\) with \(T<\infty\) such that \(u(x, t)\) is a solution of problem (1.1). As (2.8) shows,

It means that \(P(u_{0})< d\) is equivalent to \(P(u)< d\) for any \(t\in[0, T)\).

If \(u_{0}\in V\), then we have \(u\in V\) for \(t\in[0,T)\). Indeed, if it was false, there exists a first time \(t_{1}\in(0,T)\) such that \(I(u(x, t_{1})) = 0\). By (2.6), (2.7), and

we have \(u(x, t_{1}) \neq0\). Otherwise \(P(u(x, t_{1})) = 0\), which contradicts \(P(u(x, t_{1}))> 0\). From (2.9), it follows that \(P(u(x, t_{1}))\geq d\). This contradicts \(P(u(x, t)) < d\) for any \(t\in[0, T)\), since \(I(u(x, t)) < 0\). In other words, \(u(x, t) \in V\) for any \(t\in[0, T)\). So, *V* is an invariant manifold of (1.1). □

By a proof similar to that of Theorem 2.6, we can obtain the following theorem.

### Theorem 2.7

*Define*

*Then*
*W*
*is an invariant set of problem* (1.1).

## The conditions for global well-posedness

### Theorem 3.1

(Global existence)

*Let*
\(u_{0}\in W\), *then the existence time of solution*
\(u(x,t)\)
*for problem* (1.1) *is infinite*.

### Proof

If \(u_{0}\in W\), from Theorem 2.7, we know that \(u(x, t)\in W\) for \(t\in[0, T)\). For fixed \(t\in[0, T)\), we denote \(u(x, t) = u\). Then we have \(P(u) < d\), \(I(u) > 0\). It follows from (2.6) and (2.7) that

which indicates

From Proposition 2.1 and (3.1), we know that *u* globally exists on \(t \in[0,\infty)\).

Let \(u_{0} = 0\). From (2.2), we have \(u = 0\), which shows that *u* is a trivial solution of problem (1.1). □

### Lemma 3.2

*Assume that*
\(\varphi\in H\)
*and*
\(\lambda^{*}>0\)
*satisfy*
\(I(\varphi _{\lambda^{*}}) = 0\). *Suppose*
\(\lambda^{*}<1\), *then it follows that*

### Proof

From the proof of Lemma 2.5 we know

where \(a=\int(|\Delta\varphi|^{2}+|\varphi|^{2})\), \(b=\int|\varphi|^{p+2}\).

Notice that \(I(\varphi_{\lambda^{*}}) = 0\) requires

Observe that \(\varphi= \varphi_{\lambda^{*}=1}\) and \(I(\varphi) = a -b\), using (3.5), we get

and the result of Lemma 3.2 is obtained from (3.6) since \(\lambda^{*}<1\) and \(0< p<\frac {8}{n-4} \). Lemma 3.2 is proved. □

### Theorem 3.3

(Blow up in finite time)

*Let*
\(p>1\), \(n>\frac{2(p+2)}{p}\), \(u_{0}\in V\), \(E(0)< d\), *then any solution*
\(u(x, t)\)
*to problem* (1.1) *blows up in finite time*.

### Proof

Since \(u_{0}\in V\), for \(t\in[0,T)\), from Theorem 2.6 we have \(u(x,t)\in V\), *i.e.*, \(I(u)<0\). Then we obtain

Since \(u_{0}\in L^{2}(\mathbb{R}^{n})\), \(u\in L^{2}(\mathbb{R}^{n})\), by Lemma 2.2 it follows that

For fixed \(t\in[0,T)\), \(u=u(t)\). Let \(\lambda^{*}>0\) be such that

Since \(I(u)<0\), we know from Lemma 2.5 that \(\lambda^{*}<1\). Because

By Lemma 3.2, we get

By (2.8), (2.9), and \(E(0)< d\), it follows that

where \(c_{0}\) is a positive constant. Furthermore, we can get

Hence there exists a \(t_{0}\geq0\), such that \(J'(t)< J'(0)<0\) for \(t>t_{0}\) and

since \(J(0)>0\) (by \(I(u_{0})<0\)). From (3.11), we know that there exists a \(T_{1}>0\) such that \(J(t)>0\) for \(t\in[0,T_{1})\),

From (3.12), the Hölder inequality, and the Hardy inequality, we have

and it follows that

which contradicts \(T= +\infty\). Finally, we can get

*i.e.* the solution of problem (1.1) blows up in finite time. □

## References

- 1.
Karpman, VI: Lyapunov approach to the soliton stability in highly dispersive systems. I. Fourth order nonlinear Schrödinger equations. Phys. Lett. A

**215**, 254-256 (1996) - 2.
Karpman, VI, Shagalov, AG: Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion. Physica D

**144**, 194-210 (2000) - 3.
Pausader, B: Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case. Dyn. Partial Differ. Equ.

**4**, 197-225 (2007) - 4.
Fibich, G, Ilan, B, Papanicolaou, G: Self-focusing with fourth order dispersion. SIAM J. Appl. Math.

**64**, 1437-1462 (2002) - 5.
Baruch, G, Fibich, G, Mandelbaum, E: Singular solutions of the biharmonic nonlinear Schrödinger equation. SIAM J. Appl. Math.

**70**, 3319-3341 (2010) - 6.
Baruch, G, Fibich, G: Singular solutions of the L2-supercritical biharmonic nonlinear Schrödinger equation. Nonlinearity

**24**, 1843-1859 (2011) - 7.
Pausader, B, Xia, SX: Scattering theory for the fourth-order Schrödinger equation in low dimensions. Nonlinear Anal.

**26**, 2175-2191 (2013) - 8.
Wang, YZ: Nonlinear four-order Schrödinger equations with radial data. Nonlinear Anal.

**75**, 2534-2541 (2012) - 9.
Shen, JH, Yang, YB, Chen, SH, Xu, RZ: Finite time blow up of fourth-order wave equations with nonlinear strain and source terms at high energy level. Int. J. Math.

**24**, 1350043 (2013) - 10.
Zhu, SH, Yang, H, Zhang, J: Blow-up of rough solutions to the fourth-order nonlinear Schrödinger equation. Nonlinear Anal.

**74**, 6186-6201 (2011) - 11.
Zhu, SH, Zhang, J, Yang, H: Biharmonic nonlinear Schrödinger equation and the profile decomposition. Nonlinear Anal.

**74**, 6244-6255 (2011) - 12.
Liu, RX, Tian, B, Liu, LC, Qin, B, Lü, X: Bilinear forms,

*N*-soliton solution sand soliton interactions for a fourth-order dispersive nonlinear Schrödinger equation in condensed-matter physics and biophysics. Physica B**413**, 120-125 (2013) - 13.
Payne, LE, Sattinger, DH: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math.

**22**, 273-303 (1975) - 14.
Levine, HA: Instability and nonexistence of global solutions to nonlinear wave equations of the form \(Pu_{tt}=-Au+F(u)\). Trans. Am. Math. Soc.

**192**, 1-21 (1974) - 15.
Liu, YC, Zhao, JS: On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal.

**64**, 2665-2687 (2006) - 16.
Liu, YC, Xu, RZ: Potential well method for Cauchy problem of generalized double dispersion equations. J. Math. Anal. Appl.

**338**, 1169-1187 (2008) - 17.
Liu, YC, Xu, RZ: Fourth order wave equations with nonlinear strain and source terms. J. Math. Anal. Appl.

**331**, 585-607 (2007) - 18.
Xu, RZ, Liu, YC: Ill-posedness of nonlinear parabolic equation with critical initial condition. Math. Comput. Simul.

**82**, 1363-1374 (2012) - 19.
Kato, T: On nonlinear Schrödinger equations. Ann. IHP, Phys. Théor.

**46**, 113-129 (1987) - 20.
Cazenave, T: Semilinear Schrödinger Equations, vol. 10. Am. Math. Soc., Providence (2003)

- 21.
Jiang, XL, Yang, YB, Xu, RZ: Family potential wells and its applications to NLS with harmonic potential. Appl. Math. Inf. Sci.

**6**, 155-165 (2012)

## Acknowledgements

This work was supported by the National Natural Science Foundation of China (41306086), the Fundamental Research Funds for the Central Universities. Many thanks go to the reviewers for their revision suggestions, which improved the paper a lot. The authors appreciate Prof. Weike Wang for his valuable suggestions.

## Author information

## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed to the writing of this paper. XP found the motivation of this paper. YN and JL finished the proof of the main theorems and wrote the manuscript. JS and MZ provided many good ideas and assisted with writing this paper. All authors read and approved the final manuscript.

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## About this article

#### Received

#### Accepted

#### Published

#### DOI

### MSC

- 35Q55
- 35B44
- 35A01

### Keywords

- fourth-order Schrödinger equations
- blow up
- global existence