- Research
- Open Access
Global well-posedness for nonlinear fourth-order Schrödinger equations
- Xiuyan Peng^{1},
- Yi Niu^{1}Email author,
- Jie Liu^{2, 3},
- Mingyou Zhang^{1} and
- Jihong Shen^{1, 2}Email author
- Received: 12 August 2015
- Accepted: 17 January 2016
- Published: 27 January 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.
Keywords
- fourth-order Schrödinger equations
- blow up
- global existence
MSC
- 35Q55
- 35B44
- 35A01
1 Introduction
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 ∫⋅.
2 Preliminaries
Proposition 2.1
From [19, 20], we can get the following lemma.
Lemma 2.2
By a similar argument to [21], we get the following lemmas.
Lemma 2.3
Solution of (2.5) belongs to M.
Proof
Lemma 2.4
When \(\varphi(x)\in M\), we have \(d>0\).
Proof
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
Now we discuss the invariant sets of solution for problem (1.1).
Theorem 2.6
Notice that on the basis of Theorem 2.6 one says that V is an invariant set of problem (1.1).
Proof
By a proof similar to that of Theorem 2.6, we can obtain the following theorem.
Theorem 2.7
3 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
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
Proof
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
Declarations
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.
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.
Authors’ Affiliations
References
- 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) MathSciNetView ArticleMATHGoogle Scholar
- Karpman, VI, Shagalov, AG: Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion. Physica D 144, 194-210 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Pausader, B: Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case. Dyn. Partial Differ. Equ. 4, 197-225 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Fibich, G, Ilan, B, Papanicolaou, G: Self-focusing with fourth order dispersion. SIAM J. Appl. Math. 64, 1437-1462 (2002) MathSciNetMATHGoogle Scholar
- Baruch, G, Fibich, G, Mandelbaum, E: Singular solutions of the biharmonic nonlinear Schrödinger equation. SIAM J. Appl. Math. 70, 3319-3341 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Baruch, G, Fibich, G: Singular solutions of the L2-supercritical biharmonic nonlinear Schrödinger equation. Nonlinearity 24, 1843-1859 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Pausader, B, Xia, SX: Scattering theory for the fourth-order Schrödinger equation in low dimensions. Nonlinear Anal. 26, 2175-2191 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Wang, YZ: Nonlinear four-order Schrödinger equations with radial data. Nonlinear Anal. 75, 2534-2541 (2012) MathSciNetView ArticleMATHGoogle Scholar
- 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) MathSciNetView ArticleMATHGoogle Scholar
- Zhu, SH, Yang, H, Zhang, J: Blow-up of rough solutions to the fourth-order nonlinear Schrödinger equation. Nonlinear Anal. 74, 6186-6201 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Zhu, SH, Zhang, J, Yang, H: Biharmonic nonlinear Schrödinger equation and the profile decomposition. Nonlinear Anal. 74, 6244-6255 (2011) MathSciNetView ArticleMATHGoogle Scholar
- 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) View ArticleGoogle Scholar
- Payne, LE, Sattinger, DH: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 22, 273-303 (1975) MathSciNetView ArticleMATHGoogle Scholar
- 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) MATHGoogle Scholar
- Liu, YC, Zhao, JS: On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal. 64, 2665-2687 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Liu, YC, Xu, RZ: Potential well method for Cauchy problem of generalized double dispersion equations. J. Math. Anal. Appl. 338, 1169-1187 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Liu, YC, Xu, RZ: Fourth order wave equations with nonlinear strain and source terms. J. Math. Anal. Appl. 331, 585-607 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Xu, RZ, Liu, YC: Ill-posedness of nonlinear parabolic equation with critical initial condition. Math. Comput. Simul. 82, 1363-1374 (2012) View ArticleMathSciNetMATHGoogle Scholar
- Kato, T: On nonlinear Schrödinger equations. Ann. IHP, Phys. Théor. 46, 113-129 (1987) MATHGoogle Scholar
- Cazenave, T: Semilinear Schrödinger Equations, vol. 10. Am. Math. Soc., Providence (2003) MATHGoogle Scholar
- 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) MathSciNetMATHGoogle Scholar