Abstract
This paper considers the Cauchy problem of solutions for a class of sixth order 1D nonlinear wave equations at high initial energy level. By introducing a new stable set we derive the result that certain solutions with arbitrarily positive initial energy exist globally.
Keywords:
Cauchy problem; sixth order wave equation; global existence; arbitrarily positive initial energy; potential well1 Introduction
In this paper, we consider the Cauchy problem for the following 1D nonlinear wave equation of sixth order:
where , and are constants, and are given initial data, and is a given constant satisfying certain conditions to be specified later.
When Rosenau [1] was concerned with the problem of how to describe the dynamics of a dense lattice, he discovered Equation (1.1) by a continuum method. Meanwhile onedimensional homogeneous lattice wave propagation phenomena can also be described by Equation (1.1). Since then the wellposedness of Equation (1.1) have been considered by many authors, we refer the reader to [24] and the references therein.
Recently, the authors in [5] first considered the Cauchy problem for Equation (1.1). By the contraction mapping principle, they proved the existence and the uniqueness of the local solution for the Cauchy problem of Equation (1.1). By means of the potential well method, they discussed the existence and nonexistence of global solutions to this problem at the subcritical and critical initial energy level . So it is natural for us to ask what the weak solution for problem (1.1)(1.2) behaves at supcritical initial energy level . In this paper we intend to extend the existence of global solutions in [5] with arbitrarily positive initial energy. By using the potential well method [69] and introducing a new stable set we show that if the initial data satisfy some conditions, then the corresponding local weak solution with arbitrarily positive initial energy exists globally.
2 Some assumptions and preliminary lemmas
In this section we give some assumptions and preliminary results to state the main results of this paper. Throughout the present paper, just for simplicity, we denote and by and , respectively, with the norm , and the inner product .
For the Cauchy problem (1.1), (1.2) we introduce the energy functional
and the Nehari functional
Moreover we define a new stable set, which will be used to obtain the existence of a global solution with arbitrarily positive initial energy,
We show the following local existence theorem, which has been given in [5].
Theorem 2.1[5]
Suppose that, , . Then problem (1.1)(1.2) admits a unique local solutiondefined on a maximal time intervalwith. Moreover if
3 The main result and proof
In order to help the readers quickly get the main idea of the present paper, we show the main theorem in the beginning of this section.
Theorem 3.1Let, be given and let (3.1) hold. Assume that, and the initial data satisfy
Then the existence time of a global solution for problem (1.1)(1.2) is infinite.
In what follows, we show a preliminary lemma about the monotonicity of the functional , which will be used to prove the invariance of the new stable set under the flow of problem (1.1)(1.2).
Lemma 3.2Letbe given andbe the solution of Equation (1.1) with initial data. Assume thatand the initial data satisfy Equation (3.1), then the mapis strictly decreasing as long as.
Proof Let
then we get
and
Note that by testing Equation (1.1) with u we have
Then, Equation (3.4) becomes
Furthermore, from we have for . Obviously from and (3.1) we can get
which implies . It is easy to see that , namely, . Therefore, we find that the map is strictly decreasing. □
Subsequently we show the invariance of the new stable set under the flow of problem (1.1)(1.2), which plays a key role in proving existence of global solutions for problem (1.1)(1.2) at high initial energy level .
Lemma 3.3 (Invariant set)
Letbe given andbe a weak solution of problem (1.1)(1.2) with maximal existence time interval, . Assume that the initial data satisfy (3.1). Then all solutions of problem (1.1)(1.2) withbelong to, provided.
Proof We prove . If it is false, let be the first time that
namely,
Let be defined as (3.2) above. Hence by Lemma 3.2, we see that and are strictly decreasing on the interval . And then by (3.1), for all , we have
Therefore from the continuity of in t we get
On the other hand, by (2.1) and (2.2) we can obtain
Recalling (3.6) we have
Then from the following equalities:
and (3.8), we can derive
which contradicts the first inequality of (3.7). This completes the proof. □
At this point we can prove the global existence for the solution of problem (1.1)(1.2) with arbitrarily positive initial energy.
Proof of Theorem 3.1 From Theorem 2.1 there exists a unique local solution of problem (1.1)(1.2) defined on a maximal time interval . Let be the weak solution of problem (1.1)(1.2) with , and (3.1). Then from Lemma 3.3 we have , namely,
Therefore from (2.1)(2.2) and (3.6), we can obtain
which implies
Hence from Theorem 2.1 it follows that and the solution of problem (1.1)(1.2) exists globally. This completes the proof of Theorem 3.1. □
Remarks
As we all know, the global existence and finite time blowup results of solutions for problem (1.1)(1.2) at the subcritical and critical initial energy level were obtained in [5], where is the total initial energy and d is the depth of the introduced potential well. The present paper derives some sufficient conditions on the initial data such that the solution exists globally at the supcritical initial energy level . However, to the best of our knowledge there is no result as regards the blowup result of solutions with arbitrary positive initial energy for the Cauchy problem (1.1)(1.2); obviously this is an open problem. Indeed we made a try to treat this issue at the same time, however, we did not derive the invariant of an unstable set as a result of the lack of a Poincaré inequality.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The work presented here was carried out in collaboration between all authors. RZ found the motivation of this paper and suggested the outline of the proofs. JH provided many good ideas for completing this paper. YB finished the proof of the main theorem. All authors have contributed to, read and approved the manuscript.
Acknowledgements
We thank the referees for their valuable suggestions which helped us improve the paper so much. This work was supported by the National Natural Science Foundation of China (11101102), China Postdoctoral Science Foundation (2013M540270), Postdoctoral Science Foundation OF Heilongjiang Province, the Support Plan for the Young College Academic Backbone of Heilongjiang Province (1252G020), Foundational Science Foundation of Harbin Engineering University and Fundamental Research Funds for the Central Universities.
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