Abstract
A modified Novikov equation with symmetric coefficients is investigated. Provided
that the initial value
MSC: 35G25, 35L05.
Keywords:
global existence; strong solutions; blowup result1 Introduction
Many scholars have paid attention to the integrable equation
which was derived by Novikov [1]. Wellposedness of the Novikov equation in the Sobolev spaces on the torus was first
done by Tiglay in [2], and was completed on both the line and the circle by Himonas and Holliman in [3]. Its Hölder continuity properties were studied in Himonas and Holmes [4]. The periodic and the nonperiodic Cauchy problem for Eq. (1) and continuity results
for the datatosolution map in the Sobolev spaces are discussed in Grayshan [5]. A matrix Lax pair for Eq. (1) is acquired in [6] and is shown to be related to a negative flow in the SawadaKotera hierarchy. The
scattering theory is applied to find nonsmooth explicit soliton solutions with multiple
peaks for Eq. (1) in [7]. Sufficient conditions on the initial data to guarantee the formation of singularities
in finite time for Eq. (1) are given in Jiang and Ni [8]. This multiple peak property is common with the CamassaHolm and DegasperisProcesi
equations [911]. Mi and Mu [12] established many dynamic results for a modified Novikov equation with peak solution.
It is shown in Ni and Zhou [13] that the Novikov equation associated with initial value has locally wellposedness
in a Sobolev space
Observing the coefficients of the Novikov equation (1), we see that the coefficient
of
Indeed, this relationship among the coefficients plays important roles in the study of the essential dynamical properties of the Novikov model [1,2,1113]. This motivates us to study the following equation:
where
which results in the bounds of
Making use of
The objective of this work is to investigate Eq. (2). Since
The rest of this paper is organized as follows. Section 2 states the main results of this work. Section 3 proves the global existence result. The proof of a blowup result is given in Section 4.
2 Main results
We let
where
For
In order to study the existence of solutions for Eq. (2), we consider its Cauchy problem in the form
which is equivalent to
where
Theorem 1Assume that the solution of problem (3) satisfies
Theorem 2Assume that
3 Global strong solutions
For proving the global existence for problem (3), we cite the local wellposedness result presented in [18].
Lemma 3.1 ([18])
Let
Assume
with the maximal existence time
Lemma 3.2Let
Proof From Lemma 3.1, we have
Differentiating Eq. (5) with respect to x yields
which leads to
For every
It is inferred that there exists a constant
Lemma 3.3Let
where
Proof Using Eqs. (2) and (6)(8), we have
Using
Remark 1 From Lemma 3.3, we conclude that, if
Lemma 3.4If
Proof We will prove this lemma to assume
where c is a positive constant. Then
On the other hand, we have
which results in
We conclude from Eqs. (11) and (13) that
Using the first equation of system (3) one derives
from which we have the conservation law
Lemma 3.5 (Kato and Ponce [23])
If
wherecis a constant depending only onr.
Lemma 3.6 (Kato and Ponce [23])
Let
Lemma 3.7Let
For
Proof Using
Using
For
We will estimate the terms on the righthand side of Eq. (17) separately. For the first term, by using the CauchySchwartz inequality and Lemmas 3.5 and 3.6, we have
Using the above estimate to the second term yields
Using the CauchySchwartz inequality and Lemma 3.5, we obtain
For the last term in Eq. (17), using
For
For
It follows from Eqs. (18)(23) that there exists a constant c such that
Integrating both sides of the above inequality with respect to t results in inequality (16). □
Proof of Theorem 1 Using Eq. (16) with
Applying the Gronwall inequality, we get
Using Eq. (15) and Lemma 3.4, we complete the proof of Theorem 1. □
Remark 2 In fact, using
4 Proof of Theorem 2
Multiplying Eq. (2) by
When
Assume that the solution
From Eq. (27), we get
from which we derive that the
Similar to the above, we know that if
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The article is a joint work of three authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
Acknowledgements
This work is supported by both the Fundamental Research Funds for the Central Universities (JBK130401, JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).
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