The global solution and blow-up phenomena to a modified Novikov equation
© Lai et al.; licensee Springer. 2014
Received: 14 August 2013
Accepted: 23 December 2013
Published: 15 January 2014
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.
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.
which results in the bounds of for Eq. (1).
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
where . Here we note that the norms , and depend on variable t.
For and nonnegative number s, denotes the Frechet space of all continuous -valued functions on . We set .
where and are arbitrary constants. Now we give the main results for problem (3).
Theorem 2 Assume that with . If , then every solution of problem (3) exists globally in time. If , then the solution blows up in finite time if and only if becomes unbounded from below in finite time. If , then the solution blows up in finite time if and only if becomes 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 ()
Let with . Then the Cauchy problem (3) has a unique solution where depends on .
Lemma 3.2 Let , and let be the maximal existence time of the solution to problem (3). Then problem (5) has a unique solution . Moreover, the map is an increasing diffeomorphism of R with for .
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 .
It is inferred that there exists a constant such that for . It completes the proof. □
where and .
Using and solving the above equation, we complete the proof of the lemma. □
Remark 1 From Lemma 3.3, we conclude that, if , then . Since the operator preserves positivity, we get . Similarly, if , we have and .
Lemma 3.4 If , , such that (or ) and , then there exists a constant such that the solution of problem (3) satisfies .
Lemma 3.5 (Kato and Ponce )
where c is a constant depending only on r.
Lemma 3.6 (Kato and Ponce )
Integrating both sides of the above inequality with respect to t results in inequality (16). □
Using Eq. (15) and Lemma 3.4, we complete the proof of Theorem 1. □
4 Proof of Theorem 2
from which we derive that the norm of the solution to problem (3) does not blow up in finite time. From Remark 2, we know that this is impossible. Therefore, we have .
Similar to the above, we know that if , the solution of problem (3) blows up if and only if .
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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