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
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
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
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
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
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
Lemma 3.2Let
Proof Let
then we get
and
Note that by testing Equation (1.1) with u we have
Then, Equation (3.4) becomes
Furthermore, from
which implies
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)
Let
Proof We prove
namely,
Let
Therefore from the continuity of
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
where
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
Therefore from (2.1)(2.2) and (3.6), we can obtain
which implies
Hence from Theorem 2.1 it follows that
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
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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