- Open Access
A few remarks on the generalized Novikov equation
© Zhou and Chen; licensee Springer. 2013
- Received: 13 July 2013
- Accepted: 30 October 2013
- Published: 25 November 2013
This paper deals with the Cauchy problem for a generalized Novikov equation , where b is a constant. The local well-posedness in the critical Besov space is established. Moreover, a lower bound for the maximal existence time and lower semicontinuity of the existence are derived, the multi-peakon solutions are also obtained. Finally, the persistence properties in weighted spaces for the solution of this equation are considered.
MSC:35G25, 35L05, 35Q50.
- persistence properties
- local well-posedness
where b, and are arbitrary constants. Our main purpose of this paper is to establish the well-posedness in the critical Besov space and persistence in a weighted Sobolev space.
which was recently discovered by Novikov in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity . The perturbative symmetry approach  yields necessary conditions for a PDE to admit infinitely many symmetries. Using this approach, Novikov was able to isolate Eq. (1.2) and find its first few symmetries, and he subsequently found a scalar Lax pair for it, proving that the equation is integrable. By using the prolongation algebra method, Hone and Wang  gave a matrix Lax pair and many conserved densities and a bi-Hamiltonian structure of the Novikov equation, and they showed how it was related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. Then in , the authors calculated the explicit formulas for multi-peakon solutions of the Novikov equation.
Recently, by the transport equations theory and the classical Friedrichs regularization method, the authors proved that the Cauchy problem for the Novikov equation is locally well posed in the Besov spaces (with and in [5, 6], and with the critical index , in ). It was also shown in  that the Novikov equation associated with the initial value is locally well posed in the Sobolev space with by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for Eq. (1.2) were established in . A Galerkin-type approximation method was used in Himonas and Holliman’s paper  to establish the well-posedness of Novikov equation (1.2) in the Sobolev space with on both the line and the circle, and in [9, 10] the authors proved that the data-to-solution map is not globally uniformly continuous on for , this result supplements Himonas and Holliman’s works. Tiglay  showed the local well-posedness of the problem in Sobolev spaces and the existence and uniqueness of solutions for all time using orbit invariants. For analytic initial data, the existence and uniqueness of analytic solutions for Eq. (1.2) were also obtained in . Analogous to the Camassa-Holm equation, the Novikov equation possesses a blow-up phenomenon [10, 12] and global weak solutions [13, 14].
where γ, α, , and are real constants. By using Painlevé analysis in [18–20], there are only three asymptotically integrable within this family: the KdV equation, the Camassa-Holm (Eq. (1.3) with ) equation and the Degasperis-Procesi equation (Eq. (1.3) with ). The solutions of the b-equation were studied numerically for various values of b in [21, 22], where b was taken as a bifurcation parameter. The necessary conditions for integrability of the b-equation were investigated in . The b-equation also admits peakon solutions for any (see [19, 21, 22]). The well-posedness, blow-up phenomena and global solutions for the b-equation were shown in [23–25].
Recently, Mi and Mu  studied the local well-posedness in the Besov space with and . It is well known that and for any : , which shows that and are quite close, so here we first establish the local well-posedness in the critical Besov space .
Remark 1.2 Theorem 1.1 improves the corresponding result in . On the other hand, noting that the counterexample given in  cannot be applied to the case in with , the question of local well-posedness of Eq. (1.1) in remains open. Actually, this is still an open problem for the Camassa-Holm and Novikov equations.
holds for (see ), Theorem 1.1 holds true in the case of with . Besides, using similar arguments in , Theorem 1.1 can also hold true in the case of with . Furthermore, the existence of solutions to Eq. (1.1) holds as the initial data belong to with , which covers the corresponding result in .
We now get a lower bound depending only on for the maximal existence time.
Next, we shall derive lower semicontinuity of the existence time, provided the initial data is smooth enough.
then Eq. (1.1) has a unique solution .
In , the authors consider the single peakon taking the form , . Moreover, this peakon solitary is a global weak solution to Eq. (1.1). Next, we discuss the existence of multi-peakon solutions to Eq. (1.1).
In [7, 29–33], the spacial decay rates for the strong solutions to the Camassa-Holm equation, the b-equation, and the Novikov equation were established provided that the corresponding initial data decay at infinity. This kind of property is the so-called persistence property. Following the main idea of , we also prove the persistence properties in weighted spaces for the solution of Eq. (1.1). However, the hard question is that there are cubic nonlinearities in (1.1) which make the proof very difficult. First, we give the following definition of an admissible weight function.
We recall that a locally absolutely continuous function is a.e. differentiable in ℝ. Moreover, its a.e. derivative belongs to and agrees with its distributional derivative. We can now state our main result on admissible weights.
(For , one has as : the conclusion of the theorem remains true but it is not interesting in this case.) The restriction guarantees the validity of condition (1.6) for a multiplicative function .
The limit case is not covered by Theorem 1.1. The result holds true, however, for the weight with , , and , or, more generally, when . See Theorem 1.6 below, which covers the case of such fast growing weights.
- (1)Take with , and choose . In this case, Theorem 1.6 states that the conditionimplies the uniform algebraic decay in :
It is worth pointing out that this is a new result for the Novikov equation.
Choose if and if with . Such weight clearly satisfies the admissibility conditions of Definition 1.1. Applying Theorem 1.6 with , we conclude that the pointwise decay as is conserved during the evolution. Similarly, we have persistence of the decay as . Hence, our Theorem 1.6 encompasses also Theorem 4.1 of .
so that condition (1.7) is indeed weaker than condition (1.6) (see  for the details).
The plan of this paper is organized as follows. In the next section, the local well-posedness in the critical Besov space is considered and Theorem 1.1 is proved. The blow-up criteria and multi-peakon solutions are obtained in Section 3 and Theorems 1.2-1.5 are proved. In the last section, the persistence properties in weighted spaces for the solution of Eq. (1.1) are considered, and Theorems 1.6-1.7 are proved.
We can easily get the following two lemmas.
Lemma 2.1 Let . Then there exists a time such that the Cauchy problem (1.1) has a solution .
Hence Φ is Holder continuous from into .
Note that to prove in means to prove in .
According to the first step, we have that the sequence () is uniformly bounded in and tends to in , thus we can use Proposition 3 in , which implies that tends to in , i.e., for any , .
for some constant C depending only on M and b. We have completed the continuity of the map Φ in .
Now, applying to Eq. (1.1) and by the same argument to the resulting equation in terms of , we may check the continuity of the map Φ in . □
Proof of Theorem 1.1 Combining the result in Lemma 2.1 with that in Lemma 2.2, one gets the existence and uniqueness of the solution of the Cauchy problem (1.1). And Lemma 2.3 shows that the solution of the Cauchy problem (1.1) depends continuously on the initial data. This completes the proof of Theorem 1.1. □
This section is devoted to the proof of Theorems 1.2-1.5. Theorems 1.2-1.3 are based on the following lemma.
Applying Gronwall’s lemma completes the proof of (3.1).
for some universal constant . Hence Gronwall’s lemma gives inequality (3.2). □
Let be such that , where C stands for the constants used in the proof of Lemma 2.1 in . We then have a solution to Eq. (1.1) with the initial data . For the sake of uniqueness, on so that extends the solution u beyond . We conclude that and Theorem 1.2 is proved. □
Theorem 1.2 yields that . This completes the proof of Theorem 1.3. □
Therefore, is uniformly bounded in . In view of Theorem 1.2, the solution can be extended beyond . This is in conflict with the definition of . □
Remark 3.1 If , , in view of being an algebra, we have (3.10). Thus we also deduce the result of Theorem 1.4.
where , .
which leads to the conclusion of Theorem 1.5. □
For and , such weight is sub-multiplicative.
If and , then ϕ is moderate. More precisely, is -moderate for , , and .
The elementary properties of sub-multiplicative and moderate weights can be found in . Next, we prove Theorem 1.6.
Indeed, let us introduce the set , if , then ; if , then .
The constant being independent on N shows that the N-truncations of a v-moderate weight are uniformly v-moderate with respect to N.
The last factor on the right-hand side is independent of N. Now taking implies that estimate (4.5) remains valid for . □
Integrating and finally letting yields the conclusion in the case . The constants throughout the proof are independent on p. Therefore, for , one can rely on the result established for finite exponents q, and then let . The rest argument is fully similar to that of Theorem 1.6. □
The authors are very grateful to the anonymous reviewers and editors for their careful read and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by NSFC grant 11301573 and in part by the Program of Chongqing Innovation Team Project in University under Grant No. KJTD201308.
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