A modified Novikov equation with symmetric coefficients is investigated. Provided that the initial value (), does not change sign and the solution u itself belongs to , the existence and uniqueness of the global strong solutions to the equation are established in the space . A blow-up result to the development of singularities in finite time for the equation is acquired.
MSC: 35G25, 35L05.
Keywords:global existence; strong solutions; blow-up result
Many scholars have paid attention to the integrable equation
which was derived by Novikov . Well-posedness of the Novikov equation in the Sobolev spaces on the torus was first done by Tiglay in , and was completed on both the line and the circle by Himonas and Holliman in . Its Hölder continuity properties were studied in Himonas and Holmes . The periodic and the non-periodic Cauchy problem for Eq. (1) and continuity results for the data-to-solution map in the Sobolev spaces are discussed in Grayshan . A matrix Lax pair for Eq. (1) is acquired in  and is shown to be related to a negative flow in the Sawada-Kotera hierarchy. The scattering theory is applied to find non-smooth explicit soliton solutions with multiple peaks for Eq. (1) in . Sufficient conditions on the initial data to guarantee the formation of singularities in finite time for Eq. (1) are given in Jiang and Ni . This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations [9-11]. Mi and Mu  established many dynamic results for a modified Novikov equation with peak solution. It is shown in Ni and Zhou  that the Novikov equation associated with initial value has locally well-posedness in a Sobolev space with by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for Eq. (1) are established in . Using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, Yan et al. proved the global existence and blow-up phenomena for the weakly dissipative Novikov equation. For other methods to handle the Novikov equation and the related partial differential equations, the reader is referred to [15-22] and the references therein.
Indeed, this relationship among the coefficients plays important roles in the study of the essential dynamical properties of the Novikov model [1,2,11-13]. This motivates us to study the following equation:
where and are arbitrary constants. Clearly, letting and , Eq. (2) becomes the Novikov equation (1). The essential difference between Eq. (2) and the Novikov equation (1) is that Eq. (2) does not conform with the following conservation law:
The objective of this work is to investigate Eq. (2). Since and are arbitrary constants, we cannot obtain the boundedness of the solution u for Eq. (2) although the initial data satisfy the sign condition. To overcome this, assuming that the solution itself satisfies and the initial data satisfy the sign condition, we adopt the methods used in Rodriguez-Blanco  to derive that possesses bounds for any time . This leads us to establish the well-posedness of the global strong solutions to Eq. (2). Parts of the main results in [17,18] are extended. In addition, we acquire a blow-up result to the development of singularities in finite time, which includes the blow-up result in .
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 blow-up result is given in Section 4.
2 Main results
In order to study the existence of solutions for Eq. (2), we consider its Cauchy problem in the form
which is equivalent to
Theorem 2Assume thatwith. If, then every solution of problem (3) exists globally in time. If, then the solution blows up in finite time if and only ifbecomes unbounded from below in finite time. If, then the solution blows up in finite time if and only ifbecomes unbounded from above in finite time.
3 Global strong solutions
For proving the global existence for problem (3), we cite the local well-posedness result presented in .
Lemma 3.1 ()
Proof From Lemma 3.1, we have and . Thus we conclude that both functions and are bounded, Lipschitz in space and in time. Using the existence and uniqueness theorem of ordinary differential equations derives that problem (5) has a unique solution .
Differentiating Eq. (5) with respect to x yields
which leads to
Proof Using Eqs. (2) and (6)-(8), we have
where c is a positive constant. Then
On the other hand, we have
which results in
We conclude from Eqs. (11) and (13) that . To complete the proof, we use a simple density argument . Setting , we have and . Applying when , we have . □
Using the first equation of system (3) one derives
from which we have the conservation law
Lemma 3.5 (Kato and Ponce )
wherecis a constant depending only onr.
Lemma 3.6 (Kato and Ponce )
We will estimate the terms on the right-hand side of Eq. (17) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 3.5 and 3.6, we have
Using the above estimate to the second term yields
Using the Cauchy-Schwartz inequality and Lemma 3.5, we obtain
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). □
Applying the Gronwall inequality, we get
Using Eq. (15) and Lemma 3.4, we complete the proof of Theorem 1. □
4 Proof of Theorem 2
From Eq. (27), we get
The authors declare that they have no competing interests.
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.
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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