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.
Observing the coefficients of the Novikov equation (1), we see that the coefficient of is equal to the coefficient of plus the coefficient of . That is,
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:
which results in the bounds of for Eq. (1).
Making use of , , the assumption that does not change sign, and the assumption that the solution of Eq. (2) satisfies , we prove the global existence theorem of Eq. (2) in the Sobolev space,
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
We let ( ) be the space of all measurable functions h such that . We define with the standard norm . For any real number s, we let denote the Sobolev space with the norm defined by
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 .
In order to study the existence of solutions for Eq. (2), we consider its Cauchy problem in the form
which is equivalent to
where and are arbitrary constants. Now we give the main results for problem (3).
Theorem 1Assume that the solution of problem (3) satisfies and let , and for all (or equivalently for all ). Then problem (3) has a unique solution satisfying
Theorem 2Assume 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 .
Assume with . Then there exists a unique solution to problem (3) and
with the maximal existence time . First, we study the differential equation
Lemma 3.2Let , 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 ofRwith 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 .
Differentiating Eq. (5) with respect to x yields
which leads to
For every , using the Sobolev imbedding theorem yields
It is inferred that there exists a constant such that for . It completes the proof. □
Lemma 3.3Let with , and let be the maximal existence time of the problem (3). We have
where and .
Proof Using Eqs. (2) and (6)-(8), we have
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 3.4If , , such that (or ) and , then there exists a constant such that the solution of problem (3) satisfies .
Proof We will prove this lemma to assume which results in from Lemma 3.1. For , from Lemma 3.3, we have . Then does not change sign. From the assumption one derives
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 )
If , then is an algebra. Moreover
wherecis a constant depending only onr.
Lemma 3.6 (Kato and Ponce )
Let . If and , then
Lemma 3.7Let and the function is a solution of problem (3) and the initial data . Then the following results hold:
For , there is a constantconly depending onaandbsuch that
Proof Using , the Gronwall inequality and Eq. (14), one derives Eq. (15).
Using and the Parseval equality gives rise to
For , applying to both sides of the first equation of system (3) and integrating with respect to x by parts, we have the identity
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
For the last term in Eq. (17), using results in
For , it follows from Eq. (20) that
For , applying Lemma 3.5 derives
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 , we obtain
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 with , Eqs. (15) and (26), we derive that the solution of Eq. (2) in space blows up in finite time if and only if .
4 Proof of Theorem 2
Multiplying Eq. (2) by and integrating by parts, we get
When , from Eq. (27), we derive is bounded. From Lemma 3.7 and Remark 2, we see that problem (3) has a global solution in the space
Assume that the solution of problem (3) blows up in finite time in the space with . If , we assume that is bounded from below on , i.e., there exists a constant such that
From Eq. (27), we get
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 .
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).
Himonas, AA, Holliman, C: The Cauchy problem for the Novikov equation. Nonlinearity. 25, 449–479 (2012). Publisher Full Text
Grayshan, K: Peakon solutions of the Novikov equation and properties of the data-to-solution map. J. Math. Anal. Appl.. 397, 515–521 (2013). Publisher Full Text
Hone, ANW, Lundmark, H, Szmigielski, J: Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm type equation. Dyn. Partial Differ. Equ.. 6, 253–289 (2009). Publisher Full Text
Jiang, ZH, Ni, LD: Blow-up phenomenon for the integrable Novikov equation. J. Math. Anal. Appl.. 385, 551–558 (2012). Publisher Full Text
Constantin, A, Escher, J: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math.. 181, 229–243 (1998). Publisher Full Text
Mi, YS, Mu, CL: On the Cauchy problem for the modified Novikov equation with peakon solutions. J. Differ. Equ.. 254, 961–982 (2013). Publisher Full Text
Ni, L, Zhou, Y: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ.. 250, 3002–3021 (2011). Publisher Full Text
Yan, W, Li, YS, Zhang, YM: Global existence and blow-up phenomena for the weakly dissipative Novikov equation. Nonlinear Anal.. 75, 2464–2473 (2012). Publisher Full Text
Himonas, A, Misiolek, G, Ponce, G, Zhou, Y: Persistence properties and unique continuation of solutions of Camassa-Holm equation. Commun. Math. Phys.. 271, 511–522 (2007). Publisher Full Text
Rodriguez-Blanco, G: On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal.. 46, 309–327 (2001). Publisher Full Text
Lai, SY, Li, N, Wu, YH: The existence of global strong and weak solutions for the Novikov equation. J. Math. Anal. Appl.. 399, 682–691 (2013). Publisher Full Text
Lai, SY, Wu, YH: The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation. J. Differ. Equ.. 248, 2038–2063 (2010). Publisher Full Text
Yan, W, Li, YS, Zhang, YM: The Cauchy problem for the integrable Novikov equation. J. Differ. Equ.. 253, 298–318 (2012). Publisher Full Text
Zhao, L, Zhou, SG: Symbolic analysis and exact travelling wave solutions to a new modified Novikov equation. Appl. Math. Comput.. 217, 590–598 (2010). Publisher Full Text