The local strong and weak solutions to a generalized Novikov equation
© Lai and Wu; licensee Springer. 2013
Received: 8 December 2012
Accepted: 3 May 2013
Published: 20 May 2013
A nonlinear partial differential equation, which includes the Novikov equation as a special case, is investigated. The well-posedness of local strong solutions for the equation in the Sobolev space with is established. Although the -norm of the solutions to the nonlinear model does not remain constant, the existence of its local weak solutions in the lower order Sobolev space with is established under the assumptions and .
Keywordslocal strong solution local weak solution generalized Novikov equation
which has been investigated by many scholars. Grayshan  studied both the periodic and the non-periodic Cauchy problem for Eq. (1) and discussed continuity results for the data-to-solution map in the Sobolev spaces. A Galerkin-type approximation method was used in Himonas and Holliman’s paper  to establish the well-posedness of Eq. (1) in the Sobolev space with on both the line and the circle. Hone et al.  applied the scattering theory to find non-smooth explicit soliton solutions with multiple peaks for Eq. (1). This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations (see [5–7]). A matrix Lax pair for Eq. (1) was acquired in [8, 9] and was shown to be related to a negative flow in the Sawada-Kotera hierarchy. Sufficient conditions on the initial data to guarantee the formation of singularities in finite time for Eq. (1) were given in Jiang and Li . Mi and Mu  obtained many dynamic results for a modified Novikov equation with a peak solution. It is shown in Ni and Zhou  that the Novikov equation associated with the initial value is locally well-posed in 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 . Tiglay  proved the local well-posedness for the periodic Cauchy problem of the Novikov equation in Sobolev space with . The orbit invariants are used to show the existence of a periodic global strong solution if the Sobolev index and a sign condition holds. For analytic initial data, the existence and uniqueness of analytic solutions for Eq. (1) are obtained in . Using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, Yan et al.  proved that Eq. (1) is locally well-posed in the Besov space under certain assumptions. For other methods to handle the Novikov equation and the related partial differential equations, the reader is referred to [15–24] and the references therein.
which takes a key role in obtaining various dynamic properties in the previous works.
The objective of this paper is to investigate Eq. (2). Since m, a and are arbitrary constants, we do not have the result that the norm of the solution of Eq. (2) remains constant. We will apply the Kato theorem for abstract differential equations to prove the existence and uniqueness of local solutions for Eq. (2) subject to the initial value (). In addition, the existence of local weak solutions for Eq. (2) is established in the lower-order Sobolev space with under the assumptions and .
The rest of this paper is organized as follows. The main results are given in Section 2. The proof of a local well-posedness result is established in Section 3, while the existence of local weak solutions is proved in Section 4.
2 Main results
Firstly, we state some notations.
For and nonnegative number s, denotes the Frechet space of all continuous -valued functions on . We set . For simplicity, throughout this article, we let c denote any positive constant which is independent of parameter ε.
and setting with and , we know that for any , (see ).
Now, we give our main results for problem (3).
Theorem 1 Let with . Then the Cauchy problem (3) has a unique solution , where depends on .
has a unique solution , in which may depend on ε. However, we shall show that under certain assumptions, there exist two constants c and , both independent of ε, such that the solution of problem (5) satisfies for any and there exists a weak solution for problem (3). These results are summarized in the following two theorems.
Theorem 2 If with such that . Let be defined as in system (5). Then there exist two constants c and , which are independent of ε, such that the solution of problem (5) satisfies for any .
Theorem 3 Suppose that with and . Then there exists a such that problem (3) has a weak solution in the sense of distribution and .
3 Proof of Theorem 1
- (I)for with
- (II), where is bounded, uniformly on bounded sets in Y. Moreover,
- (III)extends to a map from X into X, is bounded on bounded sets in Y, and satisfies
Kato theorem (see )
We set with constant , , , , and . We know that Q is an isomorphism of onto . In order to prove Theorem 1, we only need to check that and satisfy assumptions (I)-(III).
Lemma 3.1 The operator with , belongs to .
The above three lemmas can be found in Ni and Zhou .
which completes the proof of inequality (10). □
4 Proofs of Theorems 2 and 3
Lemma 4.1 (Kato and Ponce )
where c is a constant depending only on r.
Lemma 4.2 (Kato and Ponce )
Proof Using , the Gronwall inequality and (15) derives (16).
Integrating both sides of the above inequality with respect to t results in inequality (17).
for a constant . This completes the proof of Lemma 4.3. □
Lemma 4.4 ()
where c is a constant independent of ε.
where c only depends on m, a, b.
has a unique solution . Using the theorem presented on p.51 in Li and Olver  yields that there are constants and , which are independent of ε, such that for arbitrary , which leads to the conclusion of Theorem 2. □
where , and . It follows from Aubin’s compactness theorem that there is a subsequence of , denoted by , such that and their temporal derivatives are weakly convergent to a function and its derivative in and , respectively. Moreover, for any real number , is convergent to the function u strongly in the space for and converges to strongly in the space for .
with and . Since is a separable Banach space and is a bounded sequence in the dual space of X, there exists a subsequence of , still denoted by , weakly star convergent to a function v in . As weakly converges to in , it results that almost everywhere. Thus, we obtain . □
Thanks are given to referees whose comments and suggestions are very helpful to revise our paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).
- Novikov V: Generalizations of the Camassa-Holm equation. J. Phys. A, Math. Theor. 2009., 42: Article ID 342002Google Scholar
- Grayshan K: Peakon solutions of the Novikov equation and properties of the data-to-solution map. J. Math. Anal. Appl. 2013, 397: 515–521. 10.1016/j.jmaa.2012.08.006MathSciNetView ArticleMATHGoogle Scholar
- Himonas A, Holliman C: The Cauchy problem for the Novikov equation. Nonlinearity 2012, 25: 449–479. 10.1088/0951-7715/25/2/449MathSciNetView ArticleMATHGoogle Scholar
- Hone AN, Lundmark H, Szmigielski J: Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm type equation. Dyn. Partial Differ. Equ. 2009, 6: 253–289.MathSciNetView ArticleMATHGoogle Scholar
- Camassa R, Holm D: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 1993, 71: 1661–1664. 10.1103/PhysRevLett.71.1661MathSciNetView ArticleMATHGoogle Scholar
- Constantin A, Lannes D: The hydro-dynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 2009, 193: 165–186.MathSciNetView ArticleMATHGoogle Scholar
- Constantin J, Escher J: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998, 181: 229–243. 10.1007/BF02392586MathSciNetView ArticleMATHGoogle Scholar
- Hone AN, Wang JP: Integrable peakon equations with cubic nonlinearity. J. Phys. A 2008., 41: Article ID 372002–10Google Scholar
- Himonas A, Misiolek G, Ponce G, Zhou Y: Persistence properties and unique continuation of solutions of Camassa-Holm equation. Commun. Math. Phys. 2007, 271: 511–522. 10.1007/s00220-006-0172-4MathSciNetView ArticleMATHGoogle Scholar
- Jiang ZH, Ni LD: Blow-up phenomenon for the integrable Novikov equation. J. Math. Anal. Appl. 2012, 385: 551–558. 10.1016/j.jmaa.2011.06.067MathSciNetView ArticleMATHGoogle Scholar
- Mi YS, Mu CL: On the Cauchy problem for the modified Novikov equation with peakon solutions. J. Differ. Equ. 2013, 254: 961–982. 10.1016/j.jde.2012.09.016MathSciNetView ArticleMATHGoogle Scholar
- Ni L, Zhou Y: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ. 2011, 250: 3002–3021. 10.1016/j.jde.2011.01.030MathSciNetView ArticleMATHGoogle Scholar
- Tiglay, F: The periodic Cauchy problem for Novikov’s equation. Math. AP (2010). arXiv:1009.1820v1MATHGoogle Scholar
- Yan W, Li YS, Zhang YM: The Cauchy problem for the integrable Novikov equation. J. Differ. Equ. 2012, 253: 298–318. 10.1016/j.jde.2012.03.015MathSciNetView ArticleMATHGoogle Scholar
- Yan W, Li YS, Zhang YM: Global existence and blow-up phenomena for the weakly dissipative Novikov equation. Nonlinear Anal. 2012, 75: 2464–2473. 10.1016/j.na.2011.10.044MathSciNetView ArticleMATHGoogle Scholar
- Lai SY, Li N, Wu YH: The existence of global strong and weak solutions for the Novikov equation. J. Math. Anal. Appl. 2013, 399: 682–691. 10.1016/j.jmaa.2012.10.048MathSciNetView ArticleMATHGoogle Scholar
- Lai SY, Wu YH: The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation. J. Differ. Equ. 2010, 248: 2038–2063. 10.1016/j.jde.2010.01.008MathSciNetView ArticleMATHGoogle Scholar
- Li Y, Olver P: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Equ. 2000, 162: 27–63. 10.1006/jdeq.1999.3683MathSciNetView ArticleMATHGoogle Scholar
- Mohajer K, Szmigielski J: On an inverse problem associated with an integrable equation of Camassa-Holm type: explicit formulas on the real axis. Inverse Probl. 2012., 28: Article ID 015002Google Scholar
- Yin ZY: On the Cauchy problem for an intergrable equation with peakon solutions. Ill. J. Math. 2003, 47: 649–666.MATHGoogle Scholar
- Zhao L, Zhou SG: Symbolic analysis and exact travelling wave solutions to a new modified Novikov equation. Appl. Math. Comput. 2010, 217: 590–598. 10.1016/j.amc.2010.05.093MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y: Blow-up solutions to the DGH equation. J. Funct. Anal. 2007, 250: 227–248. 10.1016/j.jfa.2007.04.019MathSciNetView ArticleMATHGoogle Scholar
- Faramarz T, Mohammad S: Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source. Bound. Value Probl. 2012., 2012: Article ID 50Google Scholar
- Zhang Y, Liu DM, Mu CL, Zheng P: Blow-up for an evolution p -Laplace system with nonlocal sources and inner absorptions. Bound. Value Probl. 2011., 2011: Article ID 29Google Scholar
- Kato T: Quasi-linear equations of evolution with applications to partial differential equations. Lecture Notes in Math. 448. In Spectral Theory and Differential Equations. Springer, Berlin; 1975:25–70.View ArticleGoogle Scholar
- Kato T, Ponce G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 1998, 41: 891–907.MathSciNetView ArticleMATHGoogle Scholar
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