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Sharp energy criteria of blow-up for the energy-critical Klein-Gordon equation
- Shihui Zhu^{1}Email author
https://doi.org/10.1186/s13660-015-0910-2
© Zhu 2015
- Received: 14 August 2015
- Accepted: 26 November 2015
- Published: 4 December 2015
Abstract
In this paper, we study the sharp energy criteria of blow-up and global existence for the nonlinear Klein-Gordon equation by the sharp Gagliardo-Nirebergy-Sobolev inequality.
Keywords
- Klein-Gordon equation
- energy-critical
- energy criteria
- blow-up
MSC
- 35Q40
- 35L05
1 Introduction
Then the question how to distinguish the domains of blow-up and global existence is of particular interest and significance for both mathematicians and physicists. Zhang [14] investigates the sharp threshold of blow-up and global existence for equation (1.1) with \(H^{1}\)-sub-critical nonlinearity (i.e. \(1< p<2^{*}-1\)) by the variational argument. We remark that the sharp threshold obtained in [14] is not the energy criteria due to the threshold is not fully determined by the \(\dot{H}^{1}\)-norm of the corresponding ground state solutions. The \(H^{1}\)-energy-critical case (i.e. \(p= 2^{*}-1\)) has not been solved.
Theorem 1.1
- (i)Ifthen the solution \(u(t,x)\) of the Cauchy problem (1.1)-(1.2) is bounded in \(H^{1}\). Moreover, \(u(t,x)\) satisfies$$ \|\nabla u_{0}\|_{L^{2}}^{2}+ \|u_{0}\|_{L^{2}}^{2}< \| \nabla W\|_{L^{2}}^{2}, $$(1.4)$$ \bigl\| \nabla u(t) \bigr\| _{L^{2}}^{2}+ \bigl\| u(t) \bigr\| _{L^{2}}^{2}< \| \nabla W\|_{L^{2}}^{2}. $$(1.5)
- (ii)
Finally, we extend this method to equation (1.1) in the \(H^{1}\)-sub-critical case: \(1< p<2^{*}-1\). The main difficulty comes from that there is no best constant of the Sobolev inequality. We use the best constant of the Gagliardo-Nirenberg inequality and some new estimates to obtain the energy inequality containing the convex property as in the \(H^{1}\)-energy-critical case. Then we can obtain the sharp energy criteria of blow-up and global existence for equation (1.1) in the \(H^{1}\)-sub-critical case. We should point out that we just consider the case \(m=1\) for simplicity, and the case \(m\neq1\) can be handled by the same argument. The method in this paper may have potential applications for nonlinear wave equations with damping term, forcing term, etc.
We conclude this section with several notations. We abbreviate \(L^{q}(\mathbb{R}^{N})\), \(\|\cdot\|_{L^{q}(\mathbb{R}^{N})}\), \(H^{1}(\mathbb{R}^{N})\), \(\dot{H}^{1}(\mathbb{R}^{N})\) and \(\int_{\mathbb{R}^{N} }\cdot \,dx\) by \(L^{q}\), \(\|\cdot\|_{q}\), \(H^{1}\), \(\dot{H}^{1}\), and \(\int\cdot \,dx\). The various positive constants will be simply denoted by C.
2 Preliminaries
In this paper, the space we work in \(H^{1}:=\{v\in L^{2} \mid \nabla v\in L^{2} \}\), is a Hilbert space. The norm of \(H^{1}\) is denoted by \(\|\cdot\|_{H^{1}}\). Ginibre et al. [2], Nakanishi [4] established the local well-posedness of the Cauchy problem (1.1)-(1.2) in energy space.
Proposition 2.1
Remark 2.2
If \(1< p< 2^{*}-1\), then according to the local well-posedness, for the solution \(u(t,x)\in C([0,T);H^{1}\times L^{2})\) of the Cauchy problem (1.1)-(1.2), we have the following alternative: either \(T=+\infty\) (global existence), or \(0< T<+\infty\) and \(\lim_{t\to T} \|u(t,x)\|_{H^{1}} =+\infty\) (blow-up).
At the end of this section, we introduce two important inequalities (see [6, 16–19]).
Lemma 2.3
Lemma 2.4
3 Main results
In this paper, the main strategy is that we will use the best constant of the Sobelev embedding inequality and the best constant of the Gagliardo-Nirenberg inequality to explore the convex properties of the energy inequality. Then, by constructing the invariant sets generated by the evolutional system, we can obtain the sharp energy criteria of blow-up and global existence for equation (1.1). Here, the energy criteria mean that the thresholds are fully expressed by the \(H^{1}\)-norm or the \(\dot{H}^{1}\)-norm of the corresponding ground state solutions.
At first, we prove Theorem 1.1, which gives the sharp energy criteria of blow-up and global existence for equation (1.1) in the \(H^{1}\)-energy-critical case.
Proof
Next, we want to extend the argument of Theorem 1.1 to equation (1.1) with the \(H^{1}\)-sub-critical nonlinearity: \(1< p<2^{*}-1\). Here, we consider the special case \(p=N=3\). There are two main difficulties: One is that if we directly use the methods of Kenig and Merle for the \(H^{1}\)-energy-critical wave and Schrödinger equations [1, 15], the best constant of the Sobolev inequality \(\|u\|_{4}\leq C(\|\nabla u\|_{2}+\| u\|_{2})\) is not determined; the other is that if we directly use Holmer and Roudenko’s arguments [20] for the \(H^{1}\)-sub-critical nonlinear Schrödinger equation, the \(L^{2}\)-norm of the solutions is not conserved. Our main strategy is to add some new estimates to the sharp Gagliardo-Nirenberg inequality. Then we can balance the \(L^{2}\)-norm of the solutions, and control the energy by the \(\| \nabla u(t)\|_{2}\). Finally, applying the convexity of the energy inequality, we establish two types of invariant evolution flows, and we obtain the sharp energy criterion of blow-up and global existence of the solutions to the Cauchy problem (1.1)-(1.2), as follows.
Theorem 3.1
- (i)
- (ii)
Proof
Remark 3.2
Theorem 3.1 can be extended to the general \(H^{1}\)-sub-critical case by the same argument. For the \(H^{1}\)-sub-critical case, we extend Holmer and Roudenko’s arguments [20] for the nonlinear Schrödinger equations to the nonlinear Klein-Gordon equations, which is one of the novelties in this paper. This argument has potential applications in the nonlinear wave equations with damping term, forcing term, etc.
It is well known that the scaling invariance brings about a lot of algebraic or geometric structures and simplifications, which are significant to analyze the nonlinear waves (see [1, 15, 17]). The nonlinear Klein-Gordon equation is lack of scaling invariance. However, one of the interesting features resulting from the breakdown of the scaling is that the sharp energy criterion of the blow-up solutions is not given by the ground state of the original nonlinear Klein-Gordon equation but that of a modified equation.
Declarations
Acknowledgements
This paper was partially done when SH Zhu visited the School of Mathematics of the Georgia Institute of Technology. SH Zhu would like to thank the hospitality of the School of Mathematics. This work was supported by the National Natural Science Foundation of China Grant No. 11371267 and Grant No. 11501395, Excellent Youth Foundation of Sichuan Scientific Committee grant No. 2014JQ0039 in China.
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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