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
MSC
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
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